user254: I find it difficult to assign a meaning to the word 'rebirth'.
Here are some hints that rebirth might not be real:

Views regarding one's past and future existence are included in the "62 false beliefs"
Those views are ascribed to non-Buddhist ascetics
Views regarding the future of the Tathagata (after death) are in the 10 or 14 "unanswered questions"
The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?
If rebirth happens that seems difficult to prove by personal experience; is it an article of faith, not something one can verify by direct experience? If so isn't that (faith instead of experience) unusual in Buddhist doctrine (isn't doctrine meant to be measurable against one's experience of the world)? Or if it is experience, what kind of experience (of other lives) is it, how are you supposed to know that so-called experience is not just a dream?
This web site (which seems to be Thai) says that rebirth is a "parable" for "simple village folks living during the time of the Buddha". It says that "Reincarnation is not a simple physical birth of a person" and "This notion of the transmigration of the soul definitely does not exist in Buddhism." The end of that page suggest that people "lower themselves into hell" or "rise to the Enlightened state of the Buddha" in this life.

I think I remember reading, sometime in the distant past, than when someone asked the Buddha about the afterlife, he replied "I'm not here to talk to you about the afterlife: I'm here to talk to you about this life."
Is it OK to believe, is it OK to say that a belief in rebirth isn't important to Buddhism? Not a big part of the historical Buddha's teaching? That when he mentioned it at all, it was to say that it didn't exist ("anatta" and "anicca"), that he didn't expect to be doing it himself, and that it wasn't worth talking about?
And/or is it a non-core part of Buddhism: something which some Buddhists believe and other Buddhists don't, a local/cultural viewpoint?
The article Two Main Schools of Buddhism says,

The areas of agreement between the two schools are as follows:

Both accept Sakyamuni Buddha as the Teacher.
The Four Noble Truths are exactly the same in both schools.
The Eightfold Path is exactly the same in both schools.
The Pattica-Samuppada or teaching on Dependent Origination is the same in both schools.
Both reject the idea of a supreme being who created and governed this world.
Both accept Anicca, Dukkha, Anatta and Sila, Samadhi, Panna without any difference.


A belief in rebirth (even Karma) isn't especially on this list.
I think I agree that the above are essential: that the historical Buddha talked about them, and that they're a necessary part of Buddhist belief.
I agree that tales of rebirth and of other lives feature in some Buddhist literature, e.g. Mahayana literature seems to have the Buddha being reborn.
I don't know a lot about Buddhism so, please, if you answer with a paraphrase of scripture, please include the name of the scripture you're quoting so that I could look it up.

user7611: Rebirth is real and eventually will be proven, as is the afterlife (with heavenly and hellish worlds).
As The Buddha explained:

"Sariputta, there are these five destinations. What are the five? Hell, the animal realm, the realm of ghosts, human beings and devas." - MN 12

"Secular Buddhism" and other modern day forms of Buddhism are heretical and fit into wrong views:

"And what is wrong view? 'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no brahmans or contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is wrong view" - MN 117 (recurs many times over and over again in the Pali canons)

Secular Buddhism and other modern day forms of Buddhism were predicted by The Buddha:

"In the same way, in the course of the future there will be monks who
  won't listen when discourses that are words of the Tathagata — deep,
  deep in their meaning, transcendent, connected with emptiness — are
  being recited. They won't lend ear, won't set their hearts on knowing
  them, won't regard these teachings as worth grasping or mastering. 
But
  they will listen when discourses that are literary works — the works
  of poets, elegant in sound, elegant in rhetoric, the work of
  outsiders, words of disciples — are recited. They will lend ear and
  set their hearts on knowing them. They will regard these teachings as
  worth grasping & mastering.
In this way the disappearance of the discourses that are words of the
  Tathagata — deep, deep in their meaning, transcendent, connected with
  emptiness — will come about." - Ani Sutta (SN 20.7)

We see that outsiders and disciples have come to replace Buddhism with new fundamentally different forms of Buddhism, so different how can they be Buddhism?
These new type of Buddhists teach people to cling to views, become attached to ideas, and use incredulity rather than trying to observe the truth.
Looks like Buddhism is almost already exterminated!
If you achieve higher states your wrong views will naturally automatically disappear and you will see that it's really true that re-birth or re-existence really is true.
The Buddha explained how many births there are:

What do you think, monks: Which is
  greater, the tears you have shed while transmigrating & wandering this
  long, long time — crying & weeping from being joined with what is
  displeasing, being separated from what is pleasing — or the water in
  the four great oceans?"
"As we understand the Dhamma taught to us by the Blessed One, this is
  the greater: the tears we have shed while transmigrating & wandering
  this long, long time — crying & weeping from being joined with what is
  displeasing, being separated from what is pleasing — not the water in
  the four great oceans."
"Excellent, monks. Excellent. It is excellent that you thus understand
  the Dhamma taught by me.
"This is the greater: the tears you have shed while transmigrating &
  wandering this long, long time — crying & weeping from being joined
  with what is displeasing, being separated from what is pleasing — not
  the water in the four great oceans.....
"Why is that? From an inconstruable beginning comes transmigration. A
  beginning point is not evident, though beings hindered by ignorance
  and fettered by craving are transmigrating & wandering on. Long have
  you thus experienced stress, experienced pain, experienced loss,
  swelling the cemeteries — enough to become disenchanted with all
  fabricated things, enough to become dispassionate, enough to be
  released." - Assu Sutta


user10712: Persistence of experience, in a continuous event during the time we are alive gives the mistaken impression that, if we pay insufficient attention, we may reach the conclusion that we who experience do not change, and are something other than the changing physical animal we are. Insight into the error of supposing an essence which sustains experience and endures beyond it is the benefit of learning the truth of the principle of Anatman or Anatta.
The desire for such a continuity and endurance may lead us to affirm it and find reasons to promise and explain it, and this is what led to doctrines of essentialism and also to the teachings of rebirth in Buddhism. Taken as a metaphor, we are reborn each and every day upon waking in the sense that, using the raw materials of memory and the rudiments of our character, we reconstruct once more reflecting on the actions (karma) attributed to our cause the glamour of being a particular person with unchanging features, when in fact what is happening is the resumption of our attachment to this narrative and the particular character it preserves.
The same metaphor allows for an interpretation of the quality of past, present, or future experiences as 'heaven(ly)' or 'hell(ish)', yet need not have anything to do with experience which occurs after we die.
In consideration of dreams versus waking experience, the former are of a different character and conform to different principles. Dreams are less enduring and rational, are dependent solely on imagination for their content, whereas the waking world is far more consistent and has a physical basis of function and endurance. Knowledge of the latter is easy to acquire and just involves careful reflection and testing. Departure from this functional basis is one of the clearest indicators of being in a dreamstate.
The expansive significance of refusing to comment on what is not currently operative (gods or demons being present or absent, death presenting a continuity or discontinuity of experience, for examples) allows us also to refuse to comment upon what The Buddha did or did not say or do in any particular event.
Outside the confines of a sangha wherein doctrines are outlined as practical platforms from which one may benefit in supposing their veracity, believing or ceasing to believe things is inconsequential, save that it informs one's own motivation and response. This will have bearing on what steps will be taken to participate in or catalyze one's own realization, and so decisions of this type can become crucial in following the Path, abandoning it, or becoming lost in delusion.
Within the foregoing it might be transparent that certain conventions are supposed effective or authentic, such as the existence of the Path or Marga, the efficacy at least in part of the sangha or congregation of aspirants in walking this Path, and most importantly, the possibility of realization and insight which might arise from walking it.
The only warnings i have repeatedly received relating to beliefs or doctrines have in fact pertained to essentiality, and were characterized as Essentialism and Nihilism, these being coupled as twin errors of extremity. Whether one's ideas adhere to main schools, if one's beliefs could be identified as core to Buddhism so-called, or even if the Four Noble Truths or Eight-Fold Path were defined differently, this seemed far less problematic than to suppose there exists a central substance from which all of reality arises which itself does not change, or to suppose that the constantly changing quality of the cosmos indicates that nothing exists at all.

user17694: Belief in rebirth is as delusional as belief in self.

user295: This is dependent of the meaning of rebirth.
Taking one meaning rebirth happens from moment to moment as one set of Kalapas pass away and a new set of Kalapas arise. In this context we can reason about it through Vipassana as it all happens within this lifetime. (http://en.wikipedia.org/wiki/Rebirth_(Buddhism)#Rebirth_as_cycle_of_consciousness)
If it takes the meaning of Patisandhi still it is subjected to the same flow. 

The bhavanga-citta is the same type of citta as the first citta in life, the patisandhi-citta (rebirth-consciousness). When the patisandhi-citta falls away it conditions the next citta to arise which is the second citta in that life. This citta is the first bhavanga-citta in life.

(source: Chapter 12 of Abhidhamma in daily life by Nina Van Gorkom)
By understanding mind matter process in this life you can understand it's working over many lives also. So the Dhamma the Buddha preached was verifiable in this life: Sandiṭṭhiko. Hence for your practice the fact that there is rebirth or not is not very relevant. From the time of birth you would have experienced stress some for or the other as per in the Dhammacakkappavattana Sutta.
Also developing Abhijñā you can verify rebirth by your self. Even this is not perfect hence can lead to wrong views. Many non Buddhist also had these abilities but this lead to wrong views (or a dilutional view in some cases).
One does not have absolute control over ones rebirth hence Annatha.

user899: In every society some children gain spontaneous recollection of their past birth. Many authors have investigated such stories and found them correct. Read 'Twenty suggestive cases of reincarnation' written by psychiatrist Ian Stevenson on the phenomena of what he calls spontaneous recall of information about previous lives by young children and some adults through hypnotism.
Read Edgar Cacy's stories about rebirth.He had discovered many stories of rebirth through self hypnosis. Rebirth is a natural phenomenon discovered by many Brahmins in India long before Load Buddha's enlightenment. Your next birth depends on the Karmic force for which the person concern is responsible for. How those man and the woman became your parents? How do some people become experts in some fields. How those people become your children? These relationships occur life after life according to the power of love and cravings. However, whether you like it or not you will be reborn some day after this life. However, no one cannot explain this philosophy in a short message like this. 

user832: Can their be rebirth or death if what one believes will be born or die turns out to be a deception?
Being under the influence of karma and this deception of identity is a result of believing in a self that one believes is an aggregate, that is born into form, that grows and changes, that dies and decomposes. Attachment is not to matter, but to a belief that we are matter or we are in matter and must get out of it, or we are somehow under the influence of matter and must get free of that.
If you have a dream that you are a child lost in the wilderness and you are starving to death, what happens to that child when you wake up? Does it go away or does it disappear because it never had any existence in the first place?
That is what karma is. Of course there can be seeds from previous interactions. I am not saying that one can just say a magic word and karma has no effect. But when we do not feed it on a daily basis, it withers like a weed that is without water.
Zen practice of Dogen diminished the role of death as of little importance. 

user471: The Buddha is like a doctor (Iti 100) who treats the illness which is suffering (dukkha). Please also see this answer.
A lay beginner who still has self view (sakkāya ditthi) i.e. someone who is not yet a stream entrant, may need a different medicine compared to someone who is more advanced.
For a beginner with self view, the Buddha prescribed the following medicine:

Taking refuge in the Buddha, Dhamma and Sangha; and adopting the five precepts as basic training rules. (AN 8.39)
For cultivation of virtue in volitional thoughts and actions, one should think (AN 5.57), "I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do".
To promote cultivation of virtue in volitional thoughts and actions, the Buddha also taught that it is a safe bet (in MN 60) to assume that one would be reborn according to one's kamma. You can also think of this as the "rebirth wager".
To reduce clinging, he (MN 143) also taught to not associate oneself with the five aggregates, but this does not apply to the thought "I am the owner of my kamma".

To someone who is more advanced or is on a higher training path, like a fully ordained monk, the Buddha taught that consciousness is not self, consciousness is dependently originated, and the SAME consciousness does not wander on, as taught in MN 38 (quoted below).

The Blessed One then asked him: “Sāti, is it true that the following
pernicious view has arisen in you: ‘As I understand the Dhamma taught
by the Blessed One, it is this same consciousness that runs and
wanders through the round of rebirths, not another’?”
“Exactly so, venerable sir. As I understand the Dhamma taught by the
Blessed One, it is this same consciousness that runs and wanders
through the round of rebirths, not another.”
“What is that consciousness, Sāti?”
“Venerable sir, it is that which speaks and feels and experiences here
and there the result of good and bad actions.”
“Misguided man, to whom have you ever known me to teach the Dhamma in
that way? Misguided man, have I not stated in many ways consciousness
to be dependently arisen, since without a condition there is no
origination of consciousness? But you, misguided man, have
misrepresented us by your wrong grasp and injured yourself and stored
up much demerit; for this will lead to your harm and suffering for a
long time.”

Usually, when people think of "rebirth", they are thinking of the SAME consciousness which "speaks and feels and experiences here and there the result of good and bad actions", as wandering on in this life and would be reborn into the next life.
In SN 12.61 (quoted below), the Buddha also talks of the mind, including consciousness, as arising as one thing and ceasing as another within a day.

“Bhikkhus, the uninstructed worldling might experience revulsion
towards this body composed of the four great elements; he might become
dispassionate towards it and be liberated from it. For what reason?
Because growth and decline is seen in this body composed of the four
great elements, it is seen being taken up and laid aside. Therefore
the uninstructed worldling might experience revulsion towards this
body composed of the four great elements; he might become
dispassionate towards it and be liberated from it.
“But, bhikkhus, as to that which is called ‘mind’ and ‘mentality’ and
‘consciousness’ —the uninstructed worldling is unable to experience
revulsion towards it, unable to become dispassionate towards it and be
liberated from it. For what reason? Because for a long time this has
been held to by him, appropriated, and grasped thus: ‘This is mine,
this I am, this is my self.’ Therefore the uninstructed worldling is
unable to experience revulsion towards it, unable to become
dispassionate towards it and be liberated from it.
“It would be better, bhikkhus, for the uninstructed worldling to take
as self this body composed of the four great elements rather than the
mind. For what reason? Because this body composed of the four great
elements is seen standing for one year, for two years, for three,
four, five, or ten years, for twenty, thirty, forty, or fifty years,
for a hundred years, or even longer. But that which is called ‘mind’
and ‘mentality’ and ‘consciousness’ arises as one thing and ceases as
another by day and by night. Just as a monkey roaming through a forest
grabs hold of one branch, lets that go and grabs another, then lets
that go and grabs still another, so too that which is called ‘mind’
and ‘mentality’ and ‘consciousness’ arises as one thing and ceases as
another by day and by night.

So, if the five aggregates of form, feeling, perception, mental formations and consciousness are dependently originated, impermanent, not-self (SN 35.85); and arising and ceasing from time to time, then who is it that lives on in this life? And who is it that gets reborn into another life? The self?
It does not mean that there is no self at all (which is nihilism or annihilationism). There is a self, but it is a mental idea (Snp 4.14) that is constructed by the five aggregates via dependent origination. The self is dependently originated, impermanent, not self (as in an eternal core of one's being, like a soul); and arising and ceasing from time to time. Just like consciousness, it is not the SAME self that wanders on within this life, and on to the next life.
Yes, there is rebirth in the suttas, but it is not rebirth of a specific person or a self or a transmigrating soul. It is the "rebirth" or continuation of suffering. Suffering has been going on for a very very long time, without a clear beginning point (SN 15.3). Suffering that continues whenever and wherever there is craving (SN 44.9). And craving appears whenever and wherever there is ignorance. Suffering does not end with physical death (which is itself suffering). Suffering is only ended by Nibbana. Nibbana is the only "place" where there is no suffering.
In my opinion, whenever the suttas speak of the recollection of past lives, it is not the recollection of the past lives of a specific person, but rather the recollection of the past lives of suffering. And whenever the suttas speak of "rebirth" or birth, it is also the "rebirth" or continuation of suffering.
Kamma and its consequences also exist. It is simply the continuity of the cause-effect chain of dependent processes. However, it's impossible to analyze how kamma works precisely (AN 4.77).

user8157: "Anatta" does not mean "no-self". It means "not-self". Therefore, the Buddha did not deny the arising of the "self" concept in the mind. In fact, the Buddha taught extensively about how the idea of "self" leads to suffering (refer to Nakulapita Sutta). 
The work "birth" ("jati") can mean the birth of the "self" idea or self-identity. In this context, you may refer to the Parileyyaka Sutta, which explains very simply. 
Clinging to experiences as "self" is inherent to the operation of karma (action) & its results (vipaka). The enlightenment of the eight fold path ends karma (refer to Nibbedhika Sutta) since there is no more "self" attachment to create karma & obsess about its results. 
The teachings of good & bad karma are called mundane or worldly teachings and are not related to enlightenment (refer to MN 117). Therefore, unenlightened people still have the belief in "self" and when they do karma, they rejoice in a personal way about the results of their good karma and suffer in a personal way about the results of their bad karma. 
Therefore, whatever the unenlightened person does, be it good or bad, "self-view" keeps arising in relation to their karma. 
This re-arising of "self-view" is one interpretation of "rebirth". 
Try this essay: ANATTĀ & REBIRTH
by Buddhadāsa Bhikkhu

user760: I know too little to have an opinion on this matter, but here is an article where Thanissaro Bhikkhu argues that rebirth is "an important working hypothesis in following the path all the way to the end of suffering", that is, that the belief in rebirth is an important part of practising Buddhism.
http://www.accesstoinsight.org/lib/authors/thanissaro/truth_of_rebirth.html

user289: I'm a secular Buddhist who takes Mahayana as my starting point. What you are discussing is the cosmology-- the idea that there are six realms, maybe one or more pure lands, and places you can go after you die.
In Glenn Wallis's book "Basic Teachings of the Buddha," I think he persuasively argued that the historical Buddha was an extinctionalist and had no use for the hindu cosmology, but everywhere he went, that is what people wanted to talk about and hear about. So he recast his basic teachings in the jargon and images of the religious beliefs around him.
As a secular believer, I have to look at the various ideas that don't square with science, or as you observe, don't square with a lot of statement you find about Buddhism, an decide  what to do with it. Is it a metaphor? Is it a useful one?
The central realization in Buddhism is that we naively think we have a permanent self (or soul) that will always be with us. One re-formulation of reincarnation to illustrate this is that, essentially every second I change and the me of a second ago is dead and gone, but there is a new one, right now. The only connection is the cause and effect of this chain of phenomena that naively looks like a permanent person or soul or self.  So every second I die and am reborn into a new person, somewhat connected to the old.
This illustrates some central Buddhist themes but breaks down when we get to the point where we die. At that point, we are just dead. We live on in our actions in the people and world that survive us, but that really is stretching the metaphor.
Another point to remember about Buddhism is that there are many version of it and within each version you generally can find a sophisticated version, focused on things like philosophical questions (do we have a permanent self, what are the consequences of the answer), mediation, maybe ethics, and there is another for lay consumption, that involves devotion, magic for this worldly benefit and a keen interest in where we go after we die.
Personally, this all works for me. I don't have to worry about where I go after I die because the thing I wanted to survive my death (my eternal self, soul or what have you) was an illusion all along. 
That said, there are many forms of Buddhism, which find it mandatory to retain a Hindu-style reincarnation, or literal heavens and hells and sometimes they have very interesting mental gymnastics to square how we can come back to life after we die or make it to a pure land, despite annata being such central theme in historical Buddhism.

user43: Rebirth is a metaphor. There is no literal rebirth (of the same person). This hope for literal rebirth is really a hidden attachment to "I", the desire to continue. Let go. It's Ok, you will die -- and never exist again. In fact, "you" do not exist even now, as a separate being. It is just wrong understanding that is called "I". The world is continuous global process of matter/energy/information transformation, with no rigid boundaries in space nor in time. The boundaries are imputed by the analytic mind, they are concepts, abstractions, simplifications. It is attachment to concepts that we liberate from in Buddhism. Attachment to concepts is what leads to fear and suffering, because the concepts always fall behind the ever changing world. When your idea of how something is "supposed to be" mismatches your idea of "how it is" -- hereby is suffering. But the world is always there, behind ideas, infinitely more nuanced and multifaceted than any conceptual model.
Rebirth is a metaphor for the continuity of information and influences. Nothing ever goes away completely. Nothing ever stays the same. Something transforms, something splits apart, something joins with something else, something pushes something. These chains of causes and effects go on like this, infinitely forking and joining. This is what rebirth is. My existence did not start when I was born, it goes way into past. My consciousness did not start when I was born. I inherited it gradually (not in one piece!) from mother and father, from books, from people, -- and they inherited it from each other and from the past. The sea of consciousness goes on like this, having its roots in the basic relationships, of fundamental patterns of the universe.
This is the part of the "rebirth" metaphor that is useful: that things are fluid and have continuity. Crude materialists assume that things completely disappear, like person disappears when it dies. But it does not disappear, it just looses its solidness, its togetherness. The knot unties, but the threads are still here. The truth is, this solidness was mostly a fiction anyway. Did you think you actually were a source of your decisions? Where do you think your desires came from, your preferences came from?
When most people think of "self" they think of some abstract core that is the subject of all experience and the agent of all actions. Buddha taught (and modern cognitive science tends to agree) that upon careful examination there is no such single core. Instead experiences result from interactions of multiple perceptory functions. Similarly, our actions are performed by a conglomerate of functions, without a single agent responsible for all choices.
When the notion of individual rebirth exists as an unexamined belief, it depends on this unexamined notion of "self" as its necessary foundation. Once practitioner is free from illusion of self (in practice, not just conceptually), the notion of individual birth/death/rebirth no longer applies. From this perspective, rebirth is a byproduct of attachment to a (substantial) self.
But because karma at large still continues to function (after Awakening), the infinitely forking/joining threads of causation that once were the subject of attachment continue to participate in activity and serve as causes for subsequent effects. In this sense, rebirth as principle goes on continuously whether we posit an individual or not.

user-1: "Views regarding one's past and future existence are included in the "62 false beliefs""
A little context: I assume the 62's mentioned above are the same as found in DN 1 (Brahmajāla Sutta) translated as "The sixty-two kinds of wrong views" by Walshe (MN 102 also develops on these wrong views).
Now, "wrong view" is understood as either "not according to the damma" or simply "not in accordance with reality". In this sutta, we read the Buddha say:

There are, monks, some ascetics and Brahmins who are speculators about the past, having fixed views about the past, and who put forward various speculative theories about the past, in eighteen different ways [...]

And he proceeds explaining the many ways in which a person may mislead him/herself into concluding something false about the past. The first, for example, is this:

Here, monks, a certain ascetic or Brahmin has by means of effort, extortion, application, earnestness and right attention attained to such a state of mental concentration that he thereby recalls past existences -- one birth, two births, ... ten births, a hundred births, a thousand births [...] Thus he remembers various past lives, their conditions and details. And he says "the self and the world are eternal, barren like a mountain-peal, set firmly as a post. There beings rush round, circulate, pass away and re-arise, but this remains eternally. Why so? I have by means of effort, extortion, attained to such a state of mental concentration that I have thereby recalled various past existences...That is how I know the self and the world are eternal"

Above he is denouncing the fragile leap of conclusion: a person remembered many past lives [and contractions and expansions of the universe] and without identifying an origin, he/she concludes there is none, "so it is eternal". Now, the buddha, when confronted with this particular question, simply declares there is no discernible beginning (SN 15:3) -- something very different from "it is eternal" and "it is not eternal".
Another example of misleading conclusion:

Here, a certain ascetic or Brahmin is a logician, a reasoner. Hammering it out by reason, following his own line of thought, he argues: "The self and the world are eternal"

In these ways, he alerts about the delicate trap of drawing wrong conclusions based on what one believes to be careful examinations (either by some experience or by some abstract reasoning) but that weren't careful enough. Moreover, he illustrates flaws using as example contemporary theories (eternalism, annihilationism, and the spectrum in between) pushed by other sects.
With these things in mind, he did not claim rebirth or views/knowledge of past existence to be false. Quite the opposite, he declares himself to have the ability to recall his past existences (and often describes episodes from them). Finally, exploring this ability is also part of an often-repeated formula for arahantship (and arahants are also described developing this ability). In MN 39, after exposing the Fourth Jhana:

"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the recollection of past lives. He recollects his manifold past lives, i.e., one birth, two births, three births, [...], many aeons of cosmic contraction, many aeons of cosmic expansion, many aeons of cosmic contraction and expansion, [recollecting], 'There I had such a name, belonged to such a clan, had such an appearance

Furthermore, he used to announce the rebirth of disciples who have died -- though not of the ones who reached the supreme goal (SN 44.9).
"Views regarding the future of the Tathagata (after death) are in the 10 or 14 "unanswered questions""
Yes. But it is important to emphasize it is "unanswered"; not a negative or positive. Even if we put aside the fact that this is a question about a Buddha, it is hard to use this to develop a trust on either view.
"The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?"
As far as I can tell, in the [pali] canon it is not said "there is no self". Rather, mostly we read instructions to observe several [conditioned] things in order to directly see "this [the thing] is not self". Additionally, we read declarations such as "All phenomena are non-self" ("sabbe dhammā anattā" -- AN 3.134).
An important point to consider is that it appears something very particular is meant by "self" which does not fit well with our common understanding of this word. In SN 22:59:

Bhikkhus, form is nonself. For if, bhikkhus, form were self, this form would not lead to affliction, and it would be possible to have it of form: 'Let my form be thus; let my form not be thus.'

And:

Is what is impermanent, suffering, and subject to change fit to be regarded thus: 'This is mine, this I am, this is myself? -- No, venerable sir.

Often times, when a challenger debates with the buddha claiming a self/eternalist view, he is debunked with questions such as "if that is so, can you change your form?". Therefore, I guess we need to, first, understand what is meant by "self" from propositions extracted from such discourses.
As for "birth" instead of "rebirth", though it's not clear to our reasoning "what re-borns" (all things above considered), the minimum takeaway of the use of this word in the discourses is that it denotes continuity.
"If rebirth happens that seems difficult to prove by personal experience"
Difficult indeed. But the texts support the possibility of personally testify it.
"is it an article of faith, not something one can verify by direct experience? If so isn't that (faith instead of experience) unusual in Buddhist doctrine (isn't doctrine meant to be measurable against one's experience of the world)?"
I guess it depends on what you mean by faith. I find it difficult to recall important claims made by the buddha that weren't suppose to be verified by one's effort in this lifetime -- so, as far as my knowledge goes, yes it would be unusual. From the canon alone, I wouldn't say this is a matter of faith in the devotional or "Belief" way, where one just take it as it is and assign a special value to it, often developing an emotional dependency towards it -- the clinging buddha frequently alerted about. 
But faith as in a trust we give to something or someone while we are applying ourselves to personally verify it, I think it fits pretty well with everything else.
"Or if it is experience, what kind of experience (of other lives) is it, how are you supposed to know that so-called experience is not just a dream?"
I honestly don't know how I would proceed to scrutinize an experience of recalling my past lives.
"is it OK to believe, is it OK to say that a belief in rebirth isn't important to Buddhism? Not a big part of the historical Buddha's teaching? That when he mentioned it at all, it was to say that it didn't exist ("anatta" and "anicca"), that he didn't expect to be doing it himself, and that it wasn't worth talking about?"
One thing we can see explicitly in the texts is that he discouraged speculations, and this subject draws speculations easily, giving its such a puzzle for reasoning and concerns the mystery of our future. Having said that, I think its safe to say it wouldn't make much sense for him to require listeners to believe and internalize "the fact of rebirth", to consider it something central to the training (if it was that important, he could have made an extra noble truth out of it :). 
There are a lot of aspects of the Dhamma that receive special attention and frequent emphasis. Urging people to believe rebirth as some sort of requirement or at face value is not something I've seen (MN 60 illustrates how the buddha approached this specific matter).
Now to the specific question: from the canon alone, rebirth is important (though, arguably not an urgent concern in the most practical sense). And, to say the least, the Buddha apparently considered worth talking about, so widespread this subject appears in the suttas, fulfilling roles therein, either as an aspect of the Dhamma, providing a broader exposition of our condition, or simply used to inspire and help us take the road to do something about it, etc. In the case of Dhamma, and this is significant, "no rebirth" does feature in wrong view.
But again, one thing is saying rebirth is important in buddhism. Other is saying (the belief of) rebirth is important to the practice. Other is saying rebirth is important to some buddhists ... 

Edited to include a few more references (thank you @Unrul3r) and provide more direct answers to the questions

254: <p>I find it difficult to assign a meaning to the word 'rebirth'.</p>

<p>Here are some hints that rebirth might not be real:</p>

<ul>
<li>Views regarding one's past and future existence are included in the "62 false beliefs"</li>
<li>Those views are ascribed to non-Buddhist ascetics</li>
<li>Views regarding the future of the Tathagata (after death) are in the 10 or 14 "unanswered questions"</li>
<li>The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?</li>
<li>If rebirth happens that seems difficult to prove by personal experience; is it an article of faith, not something one can verify by direct experience? If so isn't that (faith instead of experience) unusual in Buddhist doctrine (isn't doctrine meant to be measurable against one's experience of the world)? Or if it is experience, what kind of experience (of other lives) is it, how are you supposed to know that so-called experience is not just a dream?</li>
<li><a href="http://www.buddhanet.net/e-learning/reincarnation.htm">This web site</a> (which seems to be Thai) says that rebirth is a "parable" for "simple village folks living during the time of the Buddha". It says that "Reincarnation is not a simple physical birth of a person" and "This notion of the transmigration of the soul definitely does not exist in Buddhism." The end of that page suggest that people "lower themselves into hell" or "rise to the Enlightened state of the Buddha" <em>in this life</em>.</li>
</ul>

<p>I think I remember reading, sometime in the distant past, than when someone asked the Buddha about the afterlife, he replied "I'm not here to talk to you about the afterlife: I'm here to talk to you about <em>this</em> life."</p>

<p>Is it OK to believe, is it OK to say that a belief in rebirth isn't important to Buddhism? Not a big part of the historical Buddha's teaching? That when he mentioned it at all, it was to say that it didn't exist ("anatta" and "anicca"), that he didn't expect to be doing it himself, and that it wasn't worth talking about?</p>

<p>And/or is it a non-core part of Buddhism: something which some Buddhists believe and other Buddhists don't, a local/cultural viewpoint?</p>

<p>The article <a href="http://www.budsas.org/ebud/whatbudbeliev/59.htm">Two Main Schools of Buddhism</a> says,</p>

<blockquote>
  <p>The areas of agreement between the two schools are as follows:</p>
  
  <ul>
  <li>Both accept Sakyamuni Buddha as the Teacher.</li>
  <li>The Four Noble Truths are exactly the same in both schools.</li>
  <li>The Eightfold Path is exactly the same in both schools.</li>
  <li>The Pattica-Samuppada or teaching on Dependent Origination is the same in both schools.</li>
  <li>Both reject the idea of a supreme being who created and governed this world.</li>
  <li>Both accept Anicca, Dukkha, Anatta and Sila, Samadhi, Panna without any difference.</li>
  </ul>
</blockquote>

<p>A belief in rebirth (even Karma) isn't especially on this list.</p>

<p>I think I agree that the above are essential: that the historical Buddha talked about them, and that they're a necessary part of Buddhist belief.</p>

<p>I agree that tales of rebirth and of other lives feature in some Buddhist literature, e.g. Mahayana literature seems to have the Buddha being reborn.</p>

<p>I don't know a lot about Buddhism so, please, if you answer with a paraphrase of scripture, please include the name of the scripture you're quoting so that I could look it up.</p>
 7611: <p>Rebirth is real and eventually will be proven, as is the afterlife (with heavenly and hellish worlds).</p>

<p>As The Buddha explained:</p>

<blockquote>
  <p>"Sariputta, there are these five destinations. What are the five? Hell, the animal realm, the realm of ghosts, human beings and devas." - MN 12</p>
</blockquote>

<p>"Secular Buddhism" and other modern day forms of Buddhism are heretical and fit into wrong views:</p>

<blockquote>
  <p>"And what is wrong view? 'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no brahmans or contemplatives who, faring rightly &amp; practicing rightly, proclaim this world &amp; the next after having directly known &amp; realized it for themselves.' This is wrong view" - MN 117 (recurs many times over and over again in the Pali canons)</p>
</blockquote>

<p>Secular Buddhism and other modern day forms of Buddhism were predicted by The Buddha:</p>

<blockquote>
  <p>"In the same way, in the course of the future there will be monks who
  won't listen when discourses that are words of the Tathagata — deep,
  deep in their meaning, transcendent, connected with emptiness — are
  being recited. They won't lend ear, won't set their hearts on knowing
  them, won't regard these teachings as worth grasping or mastering. </p>
  
  <p>But
  they will listen when discourses that are literary works — the works
  of poets, elegant in sound, elegant in rhetoric, the work of
  outsiders, words of disciples — are recited. They will lend ear and
  set their hearts on knowing them. They will regard these teachings as
  worth grasping &amp; mastering.</p>
  
  <p>In this way the disappearance of the discourses that are words of the
  Tathagata — deep, deep in their meaning, transcendent, connected with
  emptiness — will come about." - Ani Sutta (SN 20.7)</p>
</blockquote>

<p>We see that outsiders and disciples have come to replace Buddhism with new fundamentally different forms of Buddhism, so different how can they be Buddhism?</p>

<p>These new type of Buddhists teach people to cling to views, become attached to ideas, and use incredulity rather than trying to observe the truth.</p>

<p>Looks like Buddhism is almost already exterminated!</p>

<p>If you achieve higher states your wrong views will naturally automatically disappear and you will see that it's really true that re-birth or re-existence really is true.</p>

<p>The Buddha explained how many births there are:</p>

<blockquote>
  <p>What do you think, monks: Which is
  greater, the tears you have shed while transmigrating &amp; wandering this
  long, long time — crying &amp; weeping from being joined with what is
  displeasing, being separated from what is pleasing — or the water in
  the four great oceans?"</p>
  
  <p>"As we understand the Dhamma taught to us by the Blessed One, this is
  the greater: the tears we have shed while transmigrating &amp; wandering
  this long, long time — crying &amp; weeping from being joined with what is
  displeasing, being separated from what is pleasing — not the water in
  the four great oceans."</p>
  
  <p>"Excellent, monks. Excellent. It is excellent that you thus understand
  the Dhamma taught by me.</p>
  
  <p>"This is the greater: the tears you have shed while transmigrating &amp;
  wandering this long, long time — crying &amp; weeping from being joined
  with what is displeasing, being separated from what is pleasing — not
  the water in the four great oceans.....</p>
  
  <p>"Why is that? From an inconstruable beginning comes transmigration. A
  beginning point is not evident, though beings hindered by ignorance
  and fettered by craving are transmigrating &amp; wandering on. Long have
  you thus experienced stress, experienced pain, experienced loss,
  swelling the cemeteries — enough to become disenchanted with all
  fabricated things, enough to become dispassionate, enough to be
  released." - Assu Sutta</p>
</blockquote>
 10712: <p>Persistence of experience, in a continuous event during the time we are alive gives the mistaken impression that, if we pay insufficient attention, we may reach the conclusion that we who experience do not change, and are something other than the changing physical animal we are. Insight into the error of supposing an essence which sustains experience and endures beyond it is the benefit of learning the truth of the principle of Anatman or Anatta.</p>

<p>The desire for such a continuity and endurance may lead us to affirm it and find reasons to promise and explain it, and this is what led to doctrines of essentialism and also to the teachings of rebirth in Buddhism. Taken as a metaphor, we are reborn each and every day upon waking in the sense that, using the raw materials of memory and the rudiments of our character, we reconstruct once more reflecting on the actions (karma) attributed to our cause the glamour of being a particular person with unchanging features, when in fact what is happening is the resumption of our attachment to this narrative and the particular character it preserves.</p>

<p>The same metaphor allows for an interpretation of the <em>quality</em> of past, present, or future experiences as 'heaven(ly)' or 'hell(ish)', yet need not have anything to do with experience which occurs after we die.</p>

<p>In consideration of dreams versus waking experience, the former are of a different character and conform to different principles. Dreams are less enduring and rational, are dependent solely on imagination for their content, whereas the waking world is far more consistent and has a physical basis of function and endurance. Knowledge of the latter is easy to acquire and just involves careful reflection and testing. Departure from this functional basis is one of the clearest indicators of being in a dreamstate.</p>

<p>The expansive significance of refusing to comment on what is not currently operative (gods or demons being present or absent, death presenting a continuity or discontinuity of experience, for examples) allows us also to refuse to comment upon what The Buddha did or did not say or do in any particular event.</p>

<p>Outside the confines of a sangha wherein doctrines are outlined as practical platforms from which one may benefit in supposing their veracity, believing or ceasing to believe things is inconsequential, save that it informs one's own motivation and response. This will have bearing on what steps will be taken to participate in or catalyze one's own realization, and so decisions of this type can become crucial in following the Path, abandoning it, or becoming lost in delusion.</p>

<p>Within the foregoing it might be transparent that certain conventions are supposed effective or authentic, such as the existence of the Path or Marga, the efficacy at least in part of the sangha or congregation of aspirants in walking this Path, and most importantly, the possibility of realization and insight which might arise from walking it.</p>

<p>The only warnings i have repeatedly received relating to beliefs or doctrines have in fact pertained to essentiality, and were characterized as Essentialism and Nihilism, these being coupled as twin errors of extremity. Whether one's ideas adhere to main schools, if one's beliefs could be identified as core to Buddhism so-called, or even if the Four Noble Truths or Eight-Fold Path were defined differently, this seemed far less problematic than to suppose there exists a central substance from which all of reality arises which itself does not change, or to suppose that the constantly changing quality of the cosmos indicates that nothing exists at all.</p>
 17694: <p>Belief in rebirth is as delusional as belief in self.</p>
 295: <p>This is dependent of the meaning of rebirth.</p>

<p>Taking one meaning rebirth happens from moment to moment as one set of Kalapas pass away and a new set of <a href="http://en.wikipedia.org/wiki/Kalapas" rel="nofollow">Kalapas</a> arise. In this context we can reason about it through Vipassana as it all happens within this lifetime. (<a href="http://en.wikipedia.org/wiki/Rebirth_(Buddhism)#Rebirth_as_cycle_of_consciousness" rel="nofollow">http://en.wikipedia.org/wiki/Rebirth_(Buddhism)#Rebirth_as_cycle_of_consciousness</a>)</p>

<p>If it takes the meaning of <a href="http://www.wisdomlib.org/definition/patisandhi/index.html" rel="nofollow">Patisandhi</a> still it is subjected to the same flow. </p>

<blockquote>
  <p>The bhavanga-citta is the same type of citta as the first citta in life, the patisandhi-citta (rebirth-consciousness). When the patisandhi-citta falls away it conditions the next citta to arise which is the second citta in that life. This citta is the first bhavanga-citta in life.</p>
</blockquote>

<p>(source: <a href="http://www.vipassana.info/nina-abhi-12.htm" rel="nofollow">Chapter 12 of Abhidhamma in daily life by Nina Van Gorkom</a>)</p>

<p>By understanding mind matter process in this life you can understand it's working over many lives also. So the <a href="http://en.wikipedia.org/wiki/Dharma_(Buddhism)" rel="nofollow">Dhamma</a> the Buddha preached was verifiable in this life: Sandiṭṭhiko. Hence for your practice the fact that there is rebirth or not is not very relevant. From the time of birth you would have experienced stress some for or the other as per in the <a href="http://en.wikipedia.org/wiki/Dhammacakkappavattana_Sutta" rel="nofollow">Dhammacakkappavattana Sutta</a>.</p>

<p>Also developing <a href="http://en.wikipedia.org/wiki/Abhij%C3%B1%C4%81" rel="nofollow">Abhijñā</a> you can verify rebirth by your self. Even this is not perfect hence can lead to wrong views. Many non Buddhist also had these abilities but this lead to wrong views (or a dilutional view in some cases).</p>

<p>One does not have absolute control over ones rebirth hence Annatha.</p>
 899: <p>In every society some children gain spontaneous recollection of their past birth. Many authors have investigated such stories and found them correct. Read 'Twenty suggestive cases of reincarnation' written by psychiatrist Ian Stevenson on the phenomena of what he calls spontaneous recall of information about previous lives by young children and some adults through hypnotism.</p>

<p>Read Edgar Cacy's stories about rebirth.He had discovered many stories of rebirth through self hypnosis. Rebirth is a natural phenomenon discovered by many Brahmins in India long before Load Buddha's enlightenment. Your next birth depends on the Karmic force for which the person concern is responsible for. How those man and the woman became your parents? How do some people become experts in some fields. How those people become your children? These relationships occur life after life according to the power of love and cravings. However, whether you like it or not you will be reborn some day after this life. However, no one cannot explain this philosophy in a short message like this. </p>
 832: <p>Can their be rebirth or death if what one believes will be born or die turns out to be a deception?
Being under the influence of karma and this deception of identity is a result of believing in a self that one believes is an aggregate, that is born into form, that grows and changes, that dies and decomposes. Attachment is not to matter, but to a belief that we are matter or we are in matter and must get out of it, or we are somehow under the influence of matter and must get free of that.
If you have a dream that you are a child lost in the wilderness and you are starving to death, what happens to that child when you wake up? Does it go away or does it disappear because it never had any existence in the first place?
That is what karma is. Of course there can be seeds from previous interactions. I am not saying that one can just say a magic word and karma has no effect. But when we do not feed it on a daily basis, it withers like a weed that is without water.
Zen practice of Dogen diminished the role of death as of little importance. </p>
 471: <p>The Buddha is like a doctor (<a href="https://www.accesstoinsight.org/tipitaka/kn/iti/iti.4.100-112.than.html#iti-100" rel="nofollow noreferrer">Iti 100</a>) who treats the illness which is suffering (<em>dukkha</em>). Please also see <a href="https://buddhism.stackexchange.com/a/45337/471">this answer</a>.</p>
<p>A lay beginner who still has self view (<em>sakkāya ditthi</em>) i.e. someone who is not yet a stream entrant, may need a different medicine compared to someone who is more advanced.</p>
<p>For a beginner with self view, the Buddha prescribed the following medicine:</p>
<ul>
<li>Taking refuge in the Buddha, Dhamma and Sangha; and adopting the five precepts as basic training rules. (<a href="https://www.accesstoinsight.org/tipitaka/an/an08/an08.039.than.html" rel="nofollow noreferrer">AN 8.39</a>)</li>
<li>For cultivation of virtue in volitional thoughts and actions, one should think (<a href="https://suttacentral.net/an5.57/en/bodhi" rel="nofollow noreferrer">AN 5.57</a>), &quot;I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do&quot;.</li>
<li>To promote cultivation of virtue in volitional thoughts and actions, the Buddha also taught that it is a safe bet (in <a href="https://www.accesstoinsight.org/tipitaka/mn/mn.060.than.html" rel="nofollow noreferrer">MN 60</a>) to assume that one would be reborn according to one's kamma. You can also think of this as the &quot;rebirth wager&quot;.</li>
<li>To reduce clinging, he (<a href="https://www.accesstoinsight.org/tipitaka/mn/mn.143.than.html" rel="nofollow noreferrer">MN 143</a>) also taught to not associate oneself with the five aggregates, but this does not apply to the thought &quot;I am the owner of my kamma&quot;.</li>
</ul>
<p>To someone who is more advanced or is on a higher training path, like a fully ordained monk, the Buddha taught that consciousness is not self, consciousness is dependently originated, and the SAME consciousness does not wander on, as taught in <a href="https://suttacentral.net/mn38/en/bodhi" rel="nofollow noreferrer">MN 38</a> (quoted below).</p>
<blockquote>
<p>The Blessed One then asked him: “Sāti, is it true that the following
pernicious view has arisen in you: ‘As I understand the Dhamma taught
by the Blessed One, it is this same consciousness that runs and
wanders through the round of rebirths, not another’?”</p>
<p>“Exactly so, venerable sir. As I understand the Dhamma taught by the
Blessed One, it is this same consciousness that runs and wanders
through the round of rebirths, not another.”</p>
<p>“What is that consciousness, Sāti?”</p>
<p>“Venerable sir, it is that which speaks and feels and experiences here
and there the result of good and bad actions.”</p>
<p>“Misguided man, to whom have you ever known me to teach the Dhamma in
that way? Misguided man, have I not stated in many ways consciousness
to be dependently arisen, since without a condition there is no
origination of consciousness? But you, misguided man, have
misrepresented us by your wrong grasp and injured yourself and stored
up much demerit; for this will lead to your harm and suffering for a
long time.”</p>
</blockquote>
<p>Usually, when people think of &quot;rebirth&quot;, they are thinking of the SAME consciousness which &quot;speaks and feels and experiences here and there the result of good and bad actions&quot;, as wandering on in this life and would be reborn into the next life.</p>
<p>In <a href="https://suttacentral.net/sn12.61/en/bodhi" rel="nofollow noreferrer">SN 12.61</a> (quoted below), the Buddha also talks of the mind, including consciousness, as arising as one thing and ceasing as another within a day.</p>
<blockquote>
<p>“Bhikkhus, the uninstructed worldling might experience revulsion
towards this body composed of the four great elements; he might become
dispassionate towards it and be liberated from it. For what reason?
Because growth and decline is seen in this body composed of the four
great elements, it is seen being taken up and laid aside. Therefore
the uninstructed worldling might experience revulsion towards this
body composed of the four great elements; he might become
dispassionate towards it and be liberated from it.</p>
<p>“But, bhikkhus, as to that which is called ‘mind’ and ‘mentality’ and
‘consciousness’ —the uninstructed worldling is unable to experience
revulsion towards it, unable to become dispassionate towards it and be
liberated from it. For what reason? Because for a long time this has
been held to by him, appropriated, and grasped thus: ‘This is mine,
this I am, this is my self.’ Therefore the uninstructed worldling is
unable to experience revulsion towards it, unable to become
dispassionate towards it and be liberated from it.</p>
<p>“It would be better, bhikkhus, for the uninstructed worldling to take
as self this body composed of the four great elements rather than the
mind. For what reason? Because this body composed of the four great
elements is seen standing for one year, for two years, for three,
four, five, or ten years, for twenty, thirty, forty, or fifty years,
for a hundred years, or even longer. But that which is called ‘mind’
and ‘mentality’ and ‘consciousness’ arises as one thing and ceases as
another by day and by night. Just as a monkey roaming through a forest
grabs hold of one branch, lets that go and grabs another, then lets
that go and grabs still another, so too that which is called ‘mind’
and ‘mentality’ and ‘consciousness’ arises as one thing and ceases as
another by day and by night.</p>
</blockquote>
<p>So, if the five aggregates of form, feeling, perception, mental formations and consciousness are dependently originated, impermanent, not-self (<a href="https://www.accesstoinsight.org/tipitaka/sn/sn35/sn35.085.than.html" rel="nofollow noreferrer">SN 35.85</a>); and arising and ceasing from time to time, then who is it that lives on in this life? And who is it that gets reborn into another life? The self?</p>
<p>It does not mean that there is no self at all (which is nihilism or annihilationism). There is a self, but it is a mental idea (<a href="https://www.accesstoinsight.org/tipitaka/kn/snp/snp.4.14.than.html" rel="nofollow noreferrer">Snp 4.14</a>) that is constructed by the five aggregates via <a href="https://buddhism.stackexchange.com/q/48/471">dependent origination</a>. The self is dependently originated, impermanent, not self (as in an eternal core of one's being, like a soul); and arising and ceasing from time to time. Just like consciousness, it is not the SAME self that wanders on within this life, and on to the next life.</p>
<p>Yes, there is rebirth in the suttas, but it is not rebirth of a specific person or a self or a transmigrating soul. It is the &quot;rebirth&quot; or continuation of suffering. Suffering has been going on for a very very long time, without a clear beginning point (<a href="https://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html" rel="nofollow noreferrer">SN 15.3</a>). Suffering that continues whenever and wherever there is craving (<a href="https://www.accesstoinsight.org/tipitaka/sn/sn44/sn44.009.than.html" rel="nofollow noreferrer">SN 44.9</a>). And craving appears whenever and wherever there is ignorance. Suffering does not end with physical death (which is itself suffering). Suffering is only ended by Nibbana. Nibbana is the only &quot;place&quot; where there is no suffering.</p>
<p>In my opinion, whenever the suttas speak of the recollection of past lives, it is not the recollection of the past lives of a specific person, but rather the recollection of the past lives of suffering. And whenever the suttas speak of &quot;rebirth&quot; or birth, it is also the &quot;rebirth&quot; or continuation of suffering.</p>
<p>Kamma and its consequences also exist. It is simply the continuity of the cause-effect chain of dependent processes. However, it's impossible to analyze how kamma works precisely (<a href="https://www.accesstoinsight.org/tipitaka/an/an04/an04.077.than.html" rel="nofollow noreferrer">AN 4.77</a>).</p>
 8157: <p>"Anatta" does not mean "no-self". It means "not-self". Therefore, the Buddha did not deny the arising of the "self" concept in the mind. In fact, the Buddha taught extensively about how the idea of "self" leads to suffering (refer to Nakulapita Sutta). </p>

<p>The work "birth" ("jati") can mean the birth of the "self" idea or self-identity. In this context, you may refer to the Parileyyaka Sutta, which explains very simply. </p>

<p>Clinging to experiences as "self" is inherent to the operation of karma (action) &amp; its results (vipaka). The enlightenment of the eight fold path ends karma (refer to Nibbedhika Sutta) since there is no more "self" attachment to create karma &amp; obsess about its results. </p>

<p>The teachings of good &amp; bad karma are called mundane or worldly teachings and are not related to enlightenment (refer to MN 117). Therefore, unenlightened people still have the belief in "self" and when they do karma, they rejoice in a personal way about the results of their good karma and suffer in a personal way about the results of their bad karma. </p>

<p>Therefore, whatever the unenlightened person does, be it good or bad, "self-view" keeps arising in relation to their karma. </p>

<p>This re-arising of "self-view" is one interpretation of "rebirth". </p>

<p>Try this essay: <a href="https://www.dhammatalks.net/Books7/Buddhadasa_Bhikkhu_Anatta_and_Rebirth.pdf" rel="nofollow noreferrer">ANATTĀ &amp; REBIRTH
by Buddhadāsa Bhikkhu</a></p>
 760: <p>I know too little to have an opinion on this matter, but here is an article where Thanissaro Bhikkhu argues that rebirth is "an important working hypothesis in following the path all the way to the end of suffering", that is, that the belief in rebirth is an important part of practising Buddhism.</p>

<p><a href="http://www.accesstoinsight.org/lib/authors/thanissaro/truth_of_rebirth.html" rel="noreferrer">http://www.accesstoinsight.org/lib/authors/thanissaro/truth_of_rebirth.html</a></p>
 289: <p>I'm a secular Buddhist who takes Mahayana as my starting point. What you are discussing is the cosmology-- the idea that there are six realms, maybe one or more pure lands, and places you can go after you die.</p>

<p>In Glenn Wallis's book "Basic Teachings of the Buddha," I think he persuasively argued that the historical Buddha was an extinctionalist and had no use for the hindu cosmology, but everywhere he went, that is what people wanted to talk about and hear about. So he recast his basic teachings in the jargon and images of the religious beliefs around him.</p>

<p>As a secular believer, I have to look at the various ideas that don't square with science, or as you observe, don't square with a lot of statement you find about Buddhism, an decide  what to do with it. Is it a metaphor? Is it a useful one?</p>

<p>The central realization in Buddhism is that we naively think we have a permanent self (or soul) that will always be with us. One re-formulation of reincarnation to illustrate this is that, essentially every second I change and the me of a second ago is dead and gone, but there is a new one, right now. The only connection is the cause and effect of this chain of phenomena that naively looks like a permanent person or soul or self.  So every second I die and am reborn into a new person, somewhat connected to the old.</p>

<p>This illustrates some central Buddhist themes but breaks down when we get to the point where we die. At that point, we are just dead. We live on in our actions in the people and world that survive us, but that really is stretching the metaphor.</p>

<p>Another point to remember about Buddhism is that there are many version of it and within each version you generally can find a sophisticated version, focused on things like philosophical questions (do we have a permanent self, what are the consequences of the answer), mediation, maybe ethics, and there is another for lay consumption, that involves devotion, magic for this worldly benefit and a keen interest in where we go after we die.</p>

<p>Personally, this all works for me. I don't have to worry about where I go after I die because the thing I wanted to survive my death (my eternal self, soul or what have you) was an illusion all along. </p>

<p>That said, there are many forms of Buddhism, which find it mandatory to retain a Hindu-style reincarnation, or literal heavens and hells and sometimes they have very interesting mental gymnastics to square how we can come back to life after we die or make it to a pure land, despite annata being such central theme in historical Buddhism.</p>
 43: <p>Rebirth is a metaphor. There is no literal rebirth (of the same person). This hope for literal rebirth is really a hidden attachment to "I", the desire to continue. Let go. It's Ok, you will die -- and never exist again. In fact, "you" do not exist even now, as a separate being. It is just wrong understanding that is called "I". The world is continuous global process of matter/energy/information transformation, with no rigid boundaries in space nor in time. The boundaries are imputed by the analytic mind, they are concepts, abstractions, simplifications. It is attachment to concepts that we liberate from in Buddhism. Attachment to concepts is what leads to fear and suffering, because the concepts always fall behind the ever changing world. When your idea of how something is "supposed to be" mismatches your idea of "how it is" -- hereby is suffering. But the world is always there, behind ideas, infinitely more nuanced and multifaceted than any conceptual model.</p>

<p>Rebirth is a metaphor for the continuity of information and influences. Nothing ever goes away completely. Nothing ever stays the same. Something transforms, something splits apart, something joins with something else, something pushes something. These chains of causes and effects go on like this, infinitely forking and joining. This is what rebirth is. My existence did not start when I was born, it goes way into past. My consciousness did not start when I was born. I inherited it gradually (not in one piece!) from mother and father, from books, from people, -- and they inherited it from each other and from the past. The sea of consciousness goes on like this, having its roots in the basic relationships, of fundamental patterns of the universe.</p>

<p>This is the part of the "rebirth" metaphor that is useful: that things are fluid and have continuity. Crude materialists assume that things completely disappear, like person disappears when it dies. But it does not disappear, it just looses its solidness, its togetherness. The knot unties, but the threads are still here. The truth is, this solidness was mostly a fiction anyway. Did you think you actually were a source of your decisions? Where do you think your desires came from, your preferences came from?</p>

<p>When most people think of "self" they think of some abstract core that is the subject of all experience and the agent of all actions. Buddha taught (and modern cognitive science tends to agree) that upon careful examination there is no such single core. Instead experiences result from interactions of multiple perceptory functions. Similarly, our actions are performed by a conglomerate of functions, without a single agent responsible for all choices.</p>

<p>When the notion of individual rebirth exists as an unexamined belief, it depends on this unexamined notion of "self" as its necessary foundation. Once practitioner is free from illusion of self (in practice, not just conceptually), the notion of individual birth/death/rebirth no longer applies. From this perspective, rebirth is a byproduct of attachment to a (substantial) self.</p>

<p>But because karma at large still continues to function (after Awakening), the infinitely forking/joining threads of causation that once were the subject of attachment continue to participate in activity and serve as causes for subsequent effects. In this sense, rebirth as principle goes on continuously whether we posit an individual or not.</p>
 -1: <p><strong>"Views regarding one's past and future existence are included in the "62 false beliefs""</strong></p>

<p>A little context: I assume the 62's mentioned above are the same as found in DN 1 (<a href="http://www.accesstoinsight.org/tipitaka/dn/dn.01.0.bodh.html" rel="noreferrer">Brahmajāla Sutta</a>) translated as "The sixty-two kinds of wrong views" by Walshe (MN 102 also develops on these wrong views).</p>

<p>Now, "wrong view" is understood as either "not according to the damma" or simply "not in accordance with reality". In this sutta, we read the Buddha say:</p>

<blockquote>
  <p>There are, monks, some ascetics and Brahmins who are speculators about the past, having fixed views about the past, and who put forward various speculative theories about the past, in eighteen different ways [...]</p>
</blockquote>

<p>And he proceeds explaining the many ways in which a person may mislead him/herself into concluding something false about the past. The first, for example, is this:</p>

<blockquote>
  <p>Here, monks, a certain ascetic or Brahmin has by means of effort, extortion, application, earnestness and right attention attained to such a state of mental concentration that he thereby recalls past existences -- one birth, two births, ... ten births, a hundred births, a thousand births [...] Thus he remembers various past lives, their conditions and details. And he says "the self and the world are eternal, barren like a mountain-peal, set firmly as a post. There beings rush round, circulate, pass away and re-arise, but this remains eternally. Why so? I have by means of effort, extortion, attained to such a state of mental concentration that I have thereby recalled various past existences...That is how I know the self and the world are eternal"</p>
</blockquote>

<p>Above he is denouncing the fragile leap of conclusion: a person remembered many past lives [and contractions and expansions of the universe] and without identifying an origin, he/she concludes there is none, "so it is eternal". Now, the buddha, when confronted with this particular question, simply declares there is no discernible beginning (SN 15:3) -- something very different from "it is eternal" and "it is not eternal".</p>

<p>Another example of misleading conclusion:</p>

<blockquote>
  <p>Here, a certain ascetic or Brahmin is a logician, a reasoner. Hammering it out by reason, following his own line of thought, he argues: "The self and the world are eternal"</p>
</blockquote>

<p>In these ways, he alerts about the delicate trap of drawing wrong conclusions based on what one believes to be careful examinations (either by some experience or by some abstract reasoning) but that weren't careful enough. Moreover, he illustrates flaws using as example contemporary theories (eternalism, annihilationism, and the spectrum in between) pushed by other sects.</p>

<p>With these things in mind, he did not claim rebirth or views/knowledge of past existence to be false. Quite the opposite, he declares himself to have the ability to recall his past existences (and often describes episodes from them). Finally, exploring this ability is also part of an often-repeated formula for arahantship (and arahants are also described developing this ability). In <a href="http://www.accesstoinsight.org/tipitaka/mn/mn.039.than.html" rel="noreferrer">MN 39</a>, after exposing the Fourth Jhana:</p>

<blockquote>
  <p>"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the recollection of past lives. He recollects his manifold past lives, i.e., one birth, two births, three births, [...], many aeons of cosmic contraction, many aeons of cosmic expansion, many aeons of cosmic contraction and expansion, [recollecting], 'There I had such a name, belonged to such a clan, had such an appearance</p>
</blockquote>

<p>Furthermore, he used to announce the rebirth of disciples who have died -- though not of the ones who reached the supreme goal <a href="http://suttacentral.net/en/sn44.9" rel="noreferrer">(SN 44.9)</a>.</p>

<p><strong>"Views regarding the future of the Tathagata (after death) are in the 10 or 14 "unanswered questions""</strong></p>

<p>Yes. But it is important to emphasize it is "unanswered"; not a negative or positive. Even if we put aside the fact that this is a question about a Buddha, it is hard to use this to develop a trust on either view.</p>

<p><strong>"The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?"</strong></p>

<p>As far as I can tell, in the [pali] canon it is not said "there is no self". Rather, mostly we read instructions to observe several [conditioned] things in order to directly see "this [the thing] is not self". Additionally, we read declarations such as "All phenomena are non-self" ("sabbe dhammā anattā" -- <a href="http://www.accesstoinsight.org/tipitaka/an/an03/an03.134.than.html" rel="noreferrer">AN 3.134</a>).</p>

<p>An important point to consider is that it appears something very particular is meant by "self" which does not fit well with our common understanding of this word. In SN 22:59:</p>

<blockquote>
  <p>Bhikkhus, form is nonself. For if, bhikkhus, form were self, this form would not lead to affliction, and it would be possible to have it of form: 'Let my form be thus; let my form not be thus.'</p>
</blockquote>

<p>And:</p>

<blockquote>
  <p>Is what is impermanent, suffering, and subject to change fit to be regarded thus: 'This is mine, this I am, this is myself? -- No, venerable sir.</p>
</blockquote>

<p>Often times, when a challenger debates with the buddha claiming a self/eternalist view, he is debunked with questions such as "if that is so, can you change your form?". Therefore, I guess we need to, first, understand what is meant by "self" from propositions extracted from such discourses.</p>

<p>As for "birth" instead of "rebirth", though it's not clear to our reasoning "what re-borns" (all things above considered), the minimum takeaway of the use of this word in the discourses is that it denotes <em>continuity</em>.</p>

<p><strong>"If rebirth happens that seems difficult to prove by personal experience"</strong></p>

<p>Difficult indeed. But the texts support the possibility of personally testify it.</p>

<p><strong>"is it an article of faith, not something one can verify by direct experience? If so isn't that (faith instead of experience) unusual in Buddhist doctrine (isn't doctrine meant to be measurable against one's experience of the world)?"</strong></p>

<p>I guess it depends on what you mean by faith. I find it difficult to recall important claims made by the buddha that weren't suppose to be verified by one's effort in this lifetime -- so, as far as my knowledge goes, yes it would be unusual. From the canon alone, I wouldn't say this is a matter of faith in the devotional or "Belief" way, where one just take it as it is and assign a special value to it, often developing an emotional dependency towards it -- the <em>clinging</em> buddha frequently alerted about. </p>

<p>But faith as in a trust we give to something or someone while we are applying ourselves to personally verify it, I think it fits pretty well with everything else.</p>

<p><strong>"Or if it is experience, what kind of experience (of other lives) is it, how are you supposed to know that so-called experience is not just a dream?"</strong></p>

<p>I honestly don't know how I would proceed to scrutinize an experience of recalling my past lives.</p>

<p><strong>"is it OK to believe, is it OK to say that a belief in rebirth isn't important to Buddhism? Not a big part of the historical Buddha's teaching? That when he mentioned it at all, it was to say that it didn't exist ("anatta" and "anicca"), that he didn't expect to be doing it himself, and that it wasn't worth talking about?"</strong></p>

<p>One thing we can see explicitly in the texts is that he discouraged speculations, and this subject draws speculations easily, giving its such a puzzle for reasoning and concerns the mystery of our future. Having said that, I think its safe to say it wouldn't make much sense for him to require listeners to believe and internalize "the fact of rebirth", to consider it something central to the training (if it was <em>that</em> important, he could have made an extra noble truth out of it :). </p>

<p>There are a lot of aspects of the Dhamma that receive special attention and frequent emphasis. Urging people to believe rebirth as some sort of requirement or at face value is not something I've seen (<a href="http://www.accesstoinsight.org/tipitaka/mn/mn.060.than.html" rel="noreferrer">MN 60</a> illustrates how the buddha approached this specific matter).</p>

<p>Now to the specific question: from the canon alone, rebirth is important (though, arguably not an urgent concern in the most practical sense). And, to say the least, the Buddha apparently considered worth talking about, so widespread this subject appears in the suttas, fulfilling roles therein, either as an aspect of the Dhamma, providing a broader exposition of our condition, or simply used to inspire and help us take the road to do something about it, etc. In the case of Dhamma, and this is significant, "no rebirth" does feature in <em>wrong view</em>.</p>

<p>But again, one thing is saying rebirth is important in buddhism. Other is saying (the belief of) rebirth is important to the practice. Other is saying rebirth is important to some buddhists ... </p>

<hr>

<p><em>Edited to include a few more references (thank you @Unrul3r) and provide more direct answers to the questions</em></p>
