Good question.

I think many physicists would say that time is humans' perception of the constant increase in entropy within any given reference frame. It began (as far as the current real universe is concerned) when the increase in entropy began and the universe left its minimum-entropy initial state - in other words, at the Big Bang. Nothing "before" the Big Bang counts as existing, as far as science is concerned, because the Big Bang acts as a barrier preventing it from having any effect on everything that happened afterward. Time ends either when a singularity is reached (i.e. you reach the center of a black hole) or, if black holes conserve information, time never ends but rather extends infinitely into the future as a ray. That said, in most of the universe time will effectively stop existing when there is no matter or energy left in the nearby area, all of it having dispersed as the universe expanded.