The human body evolved to reproduce, and then last just a while longer to help to raise the young after that. The fact that we live so long these days raises problems, because our bodies are not “designed” to do so!
There is no good evolutionary reason for a body to last very long after the reproductive age has been reached, yet we try our best to stave off death a while, by replacing the aging and dying parts of our bodies with younger, healthier parts.
A short history
The first successful transplant was in 1905, and was the cornea of an eye.
Nothing much happened after that for almost fifty years, and then after a stuttering start, medical research started producing wonder after wonder.
If you count the number of new transplant types that have been completed in each decade, the curve is unmistakable – we are fast on our way to being able to transplant virtually anything at all from one person to another. 
While organ transplantation is becoming easier over time, there is a problem with supply.
In order for you to receive a new kidney (for example), someone else must donate one of theirs. This involves finding someone with a similar body chemistry to yours, so the organ isn’t rejected, and also hoping that the person is willing to donate the kidney.
If no live donor is available, then you need to hope that someone dies to provide you with their kidney. This is a tragic thing to hope for.
A kidney is a best-case scenario, as humans have two each, so the donor can survive without it.
But if you need a heart, then the donor will most likely be dead before you get it. And you’d better hope that the donor didn’t die of heart disease!
The problem that there are simply not enough donors for each needed organ is a huge one. 
There is also a problem that organs can only survive so long outside the body, so once a donor has provided its organs, the organs must be transplanted nearly immediately, or they will die.
The current way to transport organs to the transplanting hospital is by freezing them so that decay is minimal. But even this can cause cell damage as ice crystals form and break apart the cells.
Luckily, these problems are also being solved.
Only this week, as I write, there is news of a liver-preserving machine which you can hook a liver up to. This device will then keep the liver alive, by emulating a living body. In essence, the liver does not know that it is no longer in a body, and continues functioning. 
Now that this has been done for livers, it can be expected that similar news will be announced in the next few years for almost every other organ.
The shortfall problem, that there are simply not enough donors per required organ, can be fixed with artificial organs.
Organs are generally very difficult to replace, as they do quite a number of different things. But some of the simpler organs have already been successfully replicated.
An obvious example is the heart. The first successful artificial hearts (not a pacemaker, but an actual pump) were created in the 1982. While their recipients lasted only 112 days and two years respectively after surgery, that’s still time that the patients didn’t have without the hearts.
This artificial heart design was primitive by present-day standards, but encouraged further research.
Artificial hearts are usually used as “bridges”, to keep a patient alive while waiting for a donor to supply a “real” heart. But sometimes, the artificial heart’s help gives the patients’ own heart enough rest to heal itself, and a transplant is no longer needed. 
Almost every organ can be replaced, given enough time and research.
Ears can be replaced with cochlear implants.
Eyes can be replaced, but artificial eye resolution is still very low. There are many different threads of research ongoing in this area. 
Some of the more “bag-like” organs can be very successfully replaced right now with artificial versions.
Bladders, trachea, arteries; these can all be created from stem-cells and/or plastics.
Legs and arms deserve a full chapter. There is some amazing work being done in these areas.
The most dif