Let me preface this post by saying that I absolutely, positively believe that time travel is impossible.
Having said that, I must now admit that our current laws of physics do not absolutely, positively prove that time travel is impossible, and a variety of eminent scientists over the years have tried to construct models in which time travel is allowed.
For technical reasons, physicists have looked for the possibility of "closed timelike curves." This is not as complicated as it sounds. A "timelike curve" is the path traced out through space and time by anyone traveling slower than the speed of light -- basically, all of us trace out timelike curves as we go about our daily lives. And a "closed" timelike curve means that a person returns to the exact location in space and in time. Imagine leaving your house at 7:30 am on Tuesday morning and returning home at 7:30 am on Tuesday morning. That's a closed timelike curve, and it obviously requires you to travel backwards in time.
To the best of my knowledge, the first person to suggest the possibility of closed timelike curves was Kurt Godel (of the famous incompleteness theorems in mathematics). Godel constructed a model of the universe which is rotating and nonexpanding, and he showed that this universe contained closed timelike curves. The problem is that the real universe is not rotating, and it is expanding. Back to the drawing board.
In the 1970s, Frank Tipler at Tulane revived interest in this idea by showing that if you had a rotating, very massive, infinitely long cylinder, someone traveling around the cylinder could move along a closed timelike curve. Unfortunately, the argument does not seem to work for a finite cylinder. And infinitely long cylinders are in short supply. Oops.
More recently, Kip Thorne showed that wormholes can produce closed timelike curves, and Rich Gott showed the same thing for a collection of infinitely long cosmic strings. [Full disclosure: I worked on cosmic strings myself earlier in my career. Where do you think the title of the blog came from?] But think about these proposals: a rotating universe, an infinitely long cylinder, wormholes, cosmic strings. They all have one thing in common: we've never seen any of them! That should make you suspicious. It made Stephen Hawking suspicious -- he proposed that the laws of physics really do make time travel impossible, an idea called the "Chronology Protection Conjecture."
But even if time travel were possible, scientists think that the past could not be changed, since physics would make no sense if we allowed things like the grandfather paradox. This is encoded in Igor Novikov's "Self-consistency Conjecture," which simply states what I've just said: maybe you can go back in time, but you can't stop the Kennedy assassination or make the Cubs win the 1945 World Series. Even with this assumption, you still run into the possibility of creating something from nothing. Suppose my future self built a time machine and emailed this post back in time for me to put on my blog. Then in the future, I'll just read it, copy it, and email it back in time to my past self. So who wrote it? Maybe the wildest idea of this kind was proposed by Rich Gott in this paper. He suggested that the universe itself could tunnel backwards in time, giving birth to itself! Cosmologists have never lacked for imagination.
In my mind, there's also a Fermi paradox with time travel. If visits from the future are possible, where are all the time travelers? I'd expect them to be especially common in the vicinity of major historical events, so maybe I shouldn't be too surprised that they haven't shown up at my office. Students at MIT went so far as to hold a party for future time travelers, inviting any visitors from the future to attend. No time travelers showed up. Apparently an MIT party just wasn't enough of a draw.
So even if time travel is not definitely ruled out by the laws of physics, I wouldn't bet any money on it. (Unless, of course, you happen to know the outcome in advance....)