This used to be a write-up about how the universe is bounded in time and in space, however after prompting from tdent (There is nothing logically or physically inconsistent about an infinite universe) I have radically revised the content. This is science, so our current knowledge of the universe is subject to revision. Our current (2003) knowledge is this:

The past is finite. The universe began around 13.7 billion years ago with cosmological inflation.

The future is finite. Though the universe may continue to exist forever without ever coming together in a big crunch, entropy leading to the heat death means that everything, everywhere will eventually run out of usable energy in a finite amount of time. This won't happen all at once at one time, but it is inevitable.

Space may or may not be finite. "The geometry of space-time, in the Einsteinian sense, is flat." Of course space is curved on a local scale around massive objects such as galaxies, but like the surface of boiling water in a cauldron, on a larger scale it is flat. In other words, what I said earlier about space being finite because space time curves back on itself was wrong.

So is space finite or not? I don't know man, I didn't do it. The physicists say that there are reasons to prefer an infinite universe. And yes, that does suggest that somewhere, unimaginably far away (estimated to be no further than ten to the power of (ten to the power of 118) meters away), there are other planets almost identical to this one, containing individuals indistinguishable from you.

What is certain is that only a finite part of it can affect you - the size of the region with which you can communicate or visit in a finite amount of time is finite due to the speed limit. So communicating with your doppelgänger is impossible.

The farthest you can observe is the distance that light has been able to travel during the 14 billion years since the big bang expansion began. The most distant visible objects are now about 4 X 1026 meters away--a distance that defines our observable universe, also called our Hubble volume, our horizon volume or simply our universe.
- Max Tegmark, Scientific American, May 2003

Given a finite speed and a finite time, the amount of space-time that you can reach before the heat-death of the universe is finite. Worse still, the faster you travel the less of it you see. If you always travel at close-to the speed of light, the heat-death occurs in close-to zero subjective time.

I have seen it argued that since the universe started as a point at the big bang, it cannot be infinite is size now, as nothing can grow from finite to infinite size. There are two things wrong with this:
Our understanding doesn't go all the way back to the origin. Physics as we know it breaks down before that.
The big bang describes an increase in scale, from ultra-dense to what we see now. Infinite spaces can also increase in scale.

With regards the notion that the universe is "to all intents and purposes infinite", may I remind you that infinity is infinitely larger than any very large number, and so there is an infinite difference in size between "for all intents and purposes infinite" and really infinite.

In an infinite universe, probability says that any given possible event1, even the highly improbable ones, must be happening at some place2, i.e. infinite diversity in infinite combinations.

In a non-infinite universe, not everything that can happen, does happen. The laws of probability can be used to estimate the chances of an improbable event actually occurring. Exceedingly unlikely stuff doesn't happen in a universe of finite sizes and probabilites. Not often, anyway.

1) but none of the impossible ones. Much of Science Fiction is impossible. Sorry.
2) The idea that there are real numbers that are not normal numbers suggests that some possible events may not occur even in infinite universes.

Other sources:
Time magazine article February 14 2003 on Wilkinson Microwave Anisotropy Probe (WMAP)
Scientific American May 2003 article by Max Tegmark on Parallel Universes