One quite fascinating thing in animal biology is the question of immortality or, to be more precise, the lack of it. While most unicellular organisms can divide for as long as they have the luck to find resources and space to do so, human cells can divide only a limited amount of times (approximately around 50 times, although this does not apply to stem cells that can divide an unlimited amount of times). In principle the limitation in divisions for most human cells is due to a mechanism that has been evolved and is not an intrinsic limitation. The cancer hypothesis is that the limitation makes the appearance of cancer more unlikely. If a cell is limited to just a few divisions, if it acquires a mutation that mutation is unlikely to spread to far.
The reason for this limitation are the telomeres, situated at the end of the chromosomes, that get shorter each time the cell divides. Once these telomores reach a critical size and become to small the cell will enter a state called senescence by which they will not divide again.
This is an interesting link in which they talk about this and how in the next few decades we might know enough about the effects of limited cell replication in human life expectancy, how to increase it (maybe for ever) and how to do that avoiding nasty side effects (like increased probability of dying from cancer). The website in which this is hosted is covering all sorts of news, many of them of dubious scientific interest, but the information in the link looks sound.
On the other hand in a more reliable source (PNAS) there is a nice study on how telomere dysfunction can cause genetic instability. They work on a disease known as Werner syndrome but it is quite useful stuff for cancer research. This Werner syndrome results in people aging prematurely and researchers at the Salk institute have found how extra short telomeres can be the source of the problem.