The answer to that is not entirely settled in the scientific community. The most widely accepted theory (but certainly not the only one) of the origin of the universe is the Big Bang Theory in which the universe expanded from a very hot, dense state that could be considered the 'beginning' of the past. However, there is no real consensus as to what there was before this expansion, so it is left to your imagination what existed before the Big Bang. As far as the future goes, there are two possible fates that the Big Bang predicts. Either the universe will expand to infinity, cooling as it goes, or it will begin contracting again, leading to an eventual collapse (similar to the state in which it started). In the first case, one could argue that the universe would be infinite, while the second case may be described as having a finite end. However, in the case of a collapse, it theoretically could expand again, so even that scenario might imply an infinite future.