This is something in which I have had a profound interest since I was very young. As I've grown up and read lots of books, I've realised how viable it actually is.
When someone mentions time travel, it most probably conjures up images of a person sitting in a small machine with an umbrella-type mechanism at the back of him/her (like in H.G. Well's "The Time Machine"). That fact is that it isn't like that at all. For a start it's not nearly that easy, and we are not likely to invent small, compact time machines for many centuries to come. You might be surprised to learn, however, that there are many natural time machines in the universe around today.
I'll try to explain it as best I can... You see, it's all to do with light, and how fast it moves. Near a planet like Earth, there is a sphere of gravity, which binds us and everything else to the planet's surface. It also tries, as best it can, to "pull down" the light beams. Therefore the light rays have to "fight" agains the gravitational pull, and in effect lose energy in doing so. Because of this they slow down. Now since it is the speed of light that determines the rate at which things happen, near a planet like Earth, things happen slower.
It's all very well stating the theory, but is there any proof of this? Well, a few years ago, a super-accurate clock was sent into orbit on board a satellite and it confirmed that time is running very slightly slower on Earth that it is in space.
Of cause, the time difference is only about one in a millionth part, but imagine if you could increase the gravitational field a hundred fold...or even a thousand or a million fold. The natural time machines that I mentioned earlier are such objects in space which possess such powerful gravitational fields - these objects include quasars and black holes. If a space ship came into the direct vicinity of a quasar, it would indeed experience fantastic time travel effects. The reason for this is that, although time would be running at its normal pace everywhere else in the universe, the person's personal perception of time will slow down, and so millions of years could pass by in what, to the astronomer, will only seem like a few minutes.
The only problem with this, is that it would be extremely difficult to avoid being crushed by these enormous gravitational fields, but I sincerely believe that some amazing technology in the future will solve the problem.
There is another form of time travel, but this time in the realm of the very small - in the realm of subatomic particles. You might have heard of "antimatter". Antimatter is just like normal matter, but it possesses opposite quantum properties, for instance electrical charge (for example, a proton is positive, but an antiproton is negative). But it has been proven by physicists in the 20th Century that an antiparticle travelling forward though time is mathematically equivalent to a normal particle travelling backward through time.
There are also particles called "tachyons" which, if they exist, have the unique property that they travel faster than the speed of light, which means that they travel backwards through time. This opens up the interesting possibility of actually sending messages to our ancesters. As I say, nobody really knows if they exist yet, because no particle detector has yet detected one, but it is an interesting point that they can exist. Antimatter itself was theorised to exist in a similar way, and it was later found.
Another method of travelling through time is with the use of 'wormholes'. The concept of a wormhole was first suggested by Albert Einstein and they have since been theoretically proven to exist. A wormhole is basically a 'tunnel' through hyperspace (the multi-dimentional space-time continuum). They can be used to travel anywhere in the universe instantaneously. In order to understand how they can be used to travel through time, it is important to understand a little about Einstein's Theory of Relativity.
For hundreds of years, people used to reason that since water waves need a medium on which to propagate (ie, the water!), then light waves must also need some kind of 'substance' on which to travel. (After all, it seems obvious that since they're waves, they need something to 'wave'!) Because of this, scientists speculated that there was an invisible medium which filled the entire universe. This medium was called the "ether".
But nobody had proved that the ether really existed, so two scientists decided to set up a (fairly simple) experiment to observe the effects of the planet Earth going through it. The basic idea was to test if the speed of light varied in different directions. For instance, the speed of light should be faster if the light beams were travelling in the direction of the movement of the ether relative to the Earth. To everyone's surprise, however, the speed of light was exactly the same in all directions.
This not only disproved the ether theory, but also prompted a radical rethink of science. The breakthrough came in 1905 with Einstein's Special Theory of Relativity. He said that the speed of light never changes, no matter what you take it relative to.
To clarify this, imagine a train that is going along the track at 100m/h. Say there is a motorway running right alongside the track, and a car is travelling in the same direction as the train at a speed of 70m/h. This would mean that the train is accelerating away from the car at a rate of 30m/h every hour (100 minus 70). Thus 30m/h would be the relative speed of the train with respect to the car.
If the car increased its veloctiy to 99m/h, then the train would be receding away at only 1m/h every hour (relative speed=1m/h). Obviously, if the car reaches 100m/h and exceeds that speed, then the car will begin catching up to the train.
Now imagine if we replaced the train by a beam of light. The car is at rest on the road, so the relative speed is 100m/h (light beams obviously travel a lot faster than this, to say the least, but we'll keep to the same speeds to keep it simple). The car starts its engine and begins to move. Now, you would expect the relative speed to decrease as the car was catching up to the light beam. Would it surprise you that the relative speed stays EXACTLY the same, no matter how fast the car travels?
What this essentially means, is that NOTHING can ever reach the speed of light. When Einstein thought about this, he decided that since nothing could ever "catch up to" light, then there must be something strange going on with regards to the object that is moving. He concluded that (basically) three major things took place. Firstly, the object actually reduces its length in the direction of its motion (it "flattens"). Secondly, its mass increases (because its kinetic energy adds to the amount of matter it contains, according to the equation E=mc2). Thirdly, and most importantly in our discussion, is the fact that time slows down for the object.
Imagine you went up in a spaceship, and took a round-trip around the Galaxy at about 99.99999999999999 percent of the speed of light. At this velocity, time would slow down tremendously inside the spaceship, and, effectively, for the astronaut inside it. (There is a key equation to calculate the factor by which this and other relative effects take place, which uses a variable called "beta". Basically, you divide the square of the object's velocity from the square of the speed of light, subtract the result from unity, and take the square root.) Now, despite the fact that time has slowed down for the astronaut, it is still running at its normal pace everywhere else in the universe. So, what seems like mere seconds to the astronaut, will be thousands or millions of years for the rest of humanity.
It is clear to see, then, that if you zoom off in a spaceship at a velocity very near to 186,242 miles per second (the speed of light), then you will also zoom off into the future.
I began this discussion by mentioning wormholes, and how they could be used (using the facts above) to create a permanent time-machine. It's quite simple really. If you get one end of the wormhole and accelerate it to a speed very close to that of light, then it will have travelled into the future. The other end, though, has been kept stationary. So what you end up with is a wormhole which has one end in the present and the other end in the future. If you went through the "present" end, then you would come out of the other end in the future, and when you wanted to return, you could just go back in, and travel back to the past.
It's all very well stating the theory, but, in fact, you would have a lot of difficulty SURVIVING a wormhole. The least little disturbence (never mind the entering of a large spaceship), and the wormhole simply disintegrates. Physicists have, however, theorised different ways in which this can be resolved. These involve the use of exotic matter. It comes down to the question of what types of matter can exist in the universe. What you need to keep the mouth of a wormhole open is a type with the property that it exerts negative gravity. Personally, I don't see why any form of matter can't exist, in principle, in the universe. It's a matter of a civilisation being clever enough to bring it about.
There is another little point about these relativity time-distortion effects... What relativity actually states, is that nothing can travel AT the speed of light. This isn't to say that nothings can travel FASTER than the speed of light. There is one major problem with this, though, and that is that you would need "complex" matter. If you feed a velocity value into the beta equation (detailed above) that has a value great than c (the speed of light), then you end up subtracting a value greater than one from unity, resulting in the square root of a negative value. This results in an imaginary number. An imaginary number is a number that lies outside the familiar number line - indeed on a second dimension. These are useful mathematic constructs with lots of real-world applications, but the question of whether physical objects can possess imaginary values for their properties is less than certain, and highly intriguing.
If they can, though, then as you exceed the speed of light, time slows down to such an extent that it becomes negative. In other words, it runs backwards.