user14583: 
"And what is the faculty of concentration? There is the case where a monk, a disciple of the noble ones, making it his object to let go, attains concentration, attains singleness of mind.
Indriya-vibhanga Sutta (SN 48.10)

What is exactly meant by letting go & how is this exactly done (in daily life & meditation) ?
Why is the breath not the 'object', if it's called Mindfulness of Breathing?

user8157: "Letting go" simply means to "give up craving". The Pali word is "vossagga", which means " relinquishing", "giving up", "handing over", "relaxation", "surrender", etc.  
Therefore, instead of greedily trying to focus on the breathing as an object of clinging or personal security & possessiveness, you give up this greedy selfish ambition. Instead, you sit with a completely quiet & non-ambitious mind. You "die" to all ambition. You "hand over" the meditation to the "natural" relationship & functioning of the mind & body. 
What will occur if you can do this is consciousness will automatically became aware of the breathing because the breathing, in relation to the quiet mind, becomes the most gross or coarse sense object, automatically. 
As for "anapanasati", a Pali equivalent of the English word "of" is not found in the Pali term "anapanasati". "Anapanasati" means "mindfulness with breathing". 
You did not hit the nail on the head with this question. What happened is you totally missed the boat. Because of clinging to the breathing or clinging to the bank of the river, the boat never left the shore. Instead of the current or stream of the river taking the boat to the ocean of freedom, your anchor has the boat stuck on the shore. You splash yourself with some water, as you try to row the boat frantically that is anchored to the shore, yet you believe you are a Buddhist and believe you are practising anapanasati. 
Any monk, scholar, Buddhist or other that translates "anapanasati" as "mindfulness of breathing" knows close to nothing about both Pali & practise. 
The Pali word "sati" or "mindfulness" means "to remember" and "keep in mind" and consciousness or knowing of breathing is not dependent on "memory". 
In the suttas, knowing the breathing is called "anupassi", "pajānāti", "paṭisaṃvedī", etc, and not "sati". There is no such thing as "mindfulness of breathing". 
If you are curious about why there are so few, if any, enlightened Buddhists, the ridiculous translation of "mindfulness of breathing" provides the explanation.
If you are curious why Buddhism became extinct in India, it was because of the wrong non-Buddhist yogic translation of "mindfulness of breathing", which allowed Hindu Yoga to surpass Buddhism.
If you are curious why Buddhist countries such as Sri Lanka, Thailand & Burma are so degenerate, it is because of the wrong translation "mindfulness of breathing". 
If you are curious why WW1 & WW2 occurred; or why Bolshevism took over Russia; or why the US govt did 9/11, it is because wrong translation "mindfulness of breathing". 
Because of the wrong translation of "mindfulness of breathing", people are simply not able to let go. Therefore, every evil of the world occurs because of this. 
In summary, the role of "mindfulness" or "sati" is to remember to keep the mind in a state of "letting go". When the mind is continuously in a state of "letting go" ("vossagga"), the breathing will automatically become the sense object of consciousness. 
This is why "anapanasati" means "mindfulness with breathing" rather than "mindfulness of breathing". When there is "letting go", natural automatic awareness of breathing occurs. Thus, letting go & anapanasati are inherently related. The end of the Anapanasati Sutta says: 

There is the case where a monk develops mindfulness as a factor for awakening dependent on seclusion, dependent on dispassion, dependent
  on cessation, resulting in relinquishment (vossagga).

"Letting go" is mentioned as the path in the Anapanasati Sutta but puthujjana believe "letting go" and "Anapanasati" to two different things.
This is the only way to stream-entry; to leave the bank of the river & reach the ocean, handing over the sailing to the current of the river and to the wind.

user12022: I think the context of the Sutta is important in understanding what is meant in the passage:

"And what is the faculty of concentration? There is the case where a monk, a disciple of the noble ones, making it his object to let go, attains concentration, attains singleness of mind. Quite withdrawn from sensuality, withdrawn from unskillful mental qualities, he enters & remains in the first jhana: rapture & pleasure born from withdrawal, accompanied by directed thought & evaluation. With the stilling of directed thoughts & evaluations, he enters & remains in the second jhana: rapture & pleasure born of composure, unification of awareness free from directed thought & evaluation — internal assurance. With the fading of rapture, he remains equanimous, mindful, & alert, and senses pleasure with the body. He enters & remains in the third jhana, of which the Noble Ones declare, 'Equanimous & mindful, he has a pleasant abiding.' With the abandoning of pleasure & pain — as with the earlier disappearance of elation & distress — he enters & remains in the fourth jhana: purity of equanimity & mindfulness, neither pleasure nor pain. This is called the faculty of concentration.

In a nutshell, samadhi requires that you let go of the thought-habits that prevent you from entering it, selfish self-centeredness, control freak behaviors, OCD meditation styles, as well as letting go of why we don't embrace the sila ethical behaviors. Let go, because our way of being attached to clinging and greed is in not letting it go. How to let go: deeply analyze that we are suffering because we are holding on to coming out on top, having the advantage, taking a percentage, controlling the outcome, demanding that our world serve us. Then letting it go.

user16297: Nothing wrong with B.Thanissaro's translation you quoted, but here is Bodhi's for comparison, which makes something more clear:
SN 48.9

“And what, bhikkhus, is the faculty of concentration? Here, bhikkhus,
  the noble disciple gains concentration, gains one-pointedness of mind,
  having made release (vossagga-arammanam) the object.194 
This is called the faculty of concentration.

vossagga, here is primarily referring to the release, "letting go", that is nirvana. That is, we use the samadhi faculty for the purpose of attaining nirvana. The pali grammar is ambiguous here, so it could have a secondary meaning in addition to the above, that the process of release or letting go is what we depend on to attain successively deeper levels of samadhi. But the primary meaning of that sutta passage is that the purpose of samadhi is to realize nirvana. That's what Bodhi's footnote explains the commentaries and exegetical tradition says, and in this case I agree with it, because of where vossagga appears in the 7 awakening factors formula, after "viraga nissitam, nirodha nissitam, vossagga-parinamim). Whenever you have a sequence with viraga and nirodha, the next term, whether it's vimutti, or some other word, is going to refer to the realization of nirvana.

user43: Great question, you hit the nail on the head with this question.
The goal of Buddhist path is liberation of mind. Liberation of mind is cessation of grasping and attachment. Cessation of grasping and attachment is letting go.
So when you meditate, you look at your mind, you see grasping and attachment (it is easy to detect because it always creates dukkha, the painful feeling of wrongness) - and you let go of it to gladden and liberate your mind.
Mindfulness of breathing (and of body in general) is a tool for seeing your mind, because the state of your mind is always reflected in the breathing and the body.
Starting from the most coarse and then progressively smaller stuff.
First, you let go of erratic behavior (violence, theft, adultery, intoxication, non-factual or divisive speech), then you let go of major cravings and aversions (incl. social), various conceptual attachments & stereotypes, then you let go of negative self-judgement, then let go of doubt - etc. keep going smaller and smaller until nirvana.

14583: <blockquote>
<p>&quot;And what is the faculty of concentration? There is the case where a monk, a disciple of the noble ones, making it his object to <strong>let go</strong>, attains concentration, attains singleness of mind.</p>
<p><a href="https://www.accesstoinsight.org/tipitaka/sn/sn48/sn48.010.than.html" rel="noreferrer">Indriya-vibhanga Sutta (SN 48.10)</a></p>
</blockquote>
<p>What is exactly meant by letting go &amp; how is this exactly done (in daily life &amp; meditation) ?</p>
<p>Why is the breath not the 'object', if it's called Mindfulness of Breathing?</p>
 8157: <p>"Letting go" simply means to "give up craving". The Pali word is "<a href="https://suttacentral.net/define/vossagga" rel="nofollow noreferrer">vossagga</a>", which means " relinquishing", "giving up", "handing over", "relaxation", "surrender", etc.  </p>

<p>Therefore, instead of greedily trying to focus on the breathing as an object of clinging or personal security &amp; possessiveness, you give up this greedy selfish ambition. Instead, you sit with a completely quiet &amp; non-ambitious mind. You "die" to all ambition. You "hand over" the meditation to the "natural" relationship &amp; functioning of the mind &amp; body. </p>

<p>What will occur if you can do this is consciousness will automatically became aware of the breathing because the breathing, in relation to the quiet mind, becomes the most gross or coarse sense object, automatically. </p>

<p>As for "anapanasati", a Pali equivalent of the English word "of" is not found in the Pali term "anapanasati". "Anapanasati" means "mindfulness <strong>with</strong> breathing". </p>

<p>You did not hit the nail on the head with this question. What happened is you totally missed the boat. Because of clinging to the breathing or clinging to the bank of the river, the boat never left the shore. Instead of the current or stream of the river taking the boat to the ocean of freedom, your anchor has the boat stuck on the shore. You splash yourself with some water, as you try to row the boat frantically that is anchored to the shore, yet you believe you are a Buddhist and believe you are practising anapanasati. </p>

<p>Any monk, scholar, Buddhist or other that translates "anapanasati" as "mindfulness of breathing" knows close to nothing about both Pali &amp; practise. </p>

<p>The Pali word "sati" or "mindfulness" means "to remember" and "keep in mind" and consciousness or knowing of breathing is not dependent on "memory". </p>

<p>In the suttas, knowing the breathing is called "anupassi", "pajānāti", "paṭisaṃvedī", etc, and not "sati". There is no such thing as "mindfulness of breathing". </p>

<p>If you are curious about why there are so few, if any, enlightened Buddhists, the ridiculous translation of "mindfulness of breathing" provides the explanation.</p>

<p>If you are curious why Buddhism became extinct in India, it was because of the wrong non-Buddhist yogic translation of "mindfulness of breathing", which allowed Hindu Yoga to surpass Buddhism.</p>

<p>If you are curious why Buddhist countries such as Sri Lanka, Thailand &amp; Burma are so degenerate, it is because of the wrong translation "mindfulness of breathing". </p>

<p>If you are curious why WW1 &amp; WW2 occurred; or why Bolshevism took over Russia; or why the US govt did 9/11, it is because wrong translation "mindfulness of breathing". </p>

<p>Because of the wrong translation of "mindfulness of breathing", people are simply not able to let go. Therefore, every evil of the world occurs because of this. </p>

<p>In summary, the role of "mindfulness" or "sati" is to remember to keep the mind in a state of "letting go". When the mind is continuously in a state of "letting go" ("vossagga"), the breathing will automatically become the sense object of consciousness. </p>

<p>This is why "anapanasati" means "mindfulness with breathing" rather than "mindfulness of breathing". When there is "letting go", natural automatic awareness of breathing occurs. Thus, letting go &amp; anapanasati are inherently related. The end of the Anapanasati Sutta says: </p>

<blockquote>
  <p><strong>There is the case where a monk develops mindfulness as a factor for awakening dependent on seclusion, dependent on dispassion, dependent
  on cessation, resulting in relinquishment (vossagga).</strong></p>
</blockquote>

<p>"Letting go" is mentioned as the path in the Anapanasati Sutta but puthujjana believe "letting go" and "Anapanasati" to two different things.</p>

<p>This is the only way to stream-entry; to leave the bank of the river &amp; reach the ocean, handing over the sailing to the current of the river and to the wind.</p>
 12022: <p>I think the context of the Sutta is important in understanding what is meant in the passage:</p>

<blockquote>
  <p>"And what is the faculty of concentration? There is the case where a monk, a disciple of the noble ones, making it his object to let go, attains concentration, attains singleness of mind. Quite withdrawn from sensuality, withdrawn from unskillful mental qualities, he enters &amp; remains in the first jhana: rapture &amp; pleasure born from withdrawal, accompanied by directed thought &amp; evaluation. With the stilling of directed thoughts &amp; evaluations, he enters &amp; remains in the second jhana: rapture &amp; pleasure born of composure, unification of awareness free from directed thought &amp; evaluation — internal assurance. With the fading of rapture, he remains equanimous, mindful, &amp; alert, and senses pleasure with the body. He enters &amp; remains in the third jhana, of which the Noble Ones declare, 'Equanimous &amp; mindful, he has a pleasant abiding.' With the abandoning of pleasure &amp; pain — as with the earlier disappearance of elation &amp; distress — he enters &amp; remains in the fourth jhana: purity of equanimity &amp; mindfulness, neither pleasure nor pain. This is called the faculty of concentration.</p>
</blockquote>

<p>In a nutshell, samadhi requires that you let go of the thought-habits that prevent you from entering it, selfish self-centeredness, control freak behaviors, OCD meditation styles, as well as letting go of why we don't embrace the sila ethical behaviors. Let go, because our way of being attached to clinging and greed is in not letting it go. How to let go: deeply analyze that we are suffering because we are holding on to coming out on top, having the advantage, taking a percentage, controlling the outcome, demanding that our world serve us. Then letting it go.</p>
 16297: <p>Nothing wrong with B.Thanissaro's translation you quoted, but here is Bodhi's for comparison, which makes something more clear:</p>

<p>SN 48.9</p>

<blockquote>
  <p>“And what, bhikkhus, is the faculty of concentration? Here, bhikkhus,
  the noble disciple gains concentration, gains one-pointedness of mind,
  having made <strong>release</strong> (vossagga-arammanam) the object.194 </p>
  
  <p>This is called the faculty of concentration.</p>
</blockquote>

<p>vossagga, here is primarily referring to the release, "letting go", that is nirvana. That is, we use the samadhi faculty for the purpose of attaining nirvana. The pali grammar is ambiguous here, so it could have a secondary meaning in addition to the above, that the process of release or letting go is what we depend on to attain successively deeper levels of samadhi. But the primary meaning of that sutta passage is that the purpose of samadhi is to realize nirvana. That's what Bodhi's footnote explains the commentaries and exegetical tradition says, and in this case I agree with it, because of where vossagga appears in the 7 awakening factors formula, after "viraga nissitam, nirodha nissitam, vossagga-parinamim). Whenever you have a sequence with viraga and nirodha, the next term, whether it's vimutti, or some other word, is going to refer to the realization of nirvana.</p>
 43: <p>Great question, you hit the nail on the head with this question.</p>

<p>The goal of Buddhist path is liberation of mind. Liberation of mind is cessation of grasping and attachment. Cessation of grasping and attachment is letting go.</p>

<p>So when you meditate, you look at your mind, you see grasping and attachment (it is easy to detect because it always creates dukkha, the painful feeling of wrongness) - and you let go of it to gladden and liberate your mind.</p>

<p>Mindfulness of breathing (and of body in general) is a tool for seeing your mind, because the state of your mind is always reflected in the breathing and the body.</p>

<p>Starting from the most coarse and then progressively smaller stuff.</p>

<p>First, you let go of erratic behavior (violence, theft, adultery, intoxication, non-factual or divisive speech), then you let go of major cravings and aversions (incl. social), various conceptual attachments &amp; stereotypes, then you let go of negative self-judgement, then let go of doubt - etc. keep going smaller and smaller until nirvana.</p>
