Part 0: More historical background
Part 1: The problem
Part 2: Space-Time
Ok, I'm going to go off on another tangent, but I promise it will be interesting.
One thing that SR necessarily implies is that if you move faster than light, you can travel backwards in time. In the diagram below (see part 2 for an explanation of space-time diagrams), we are considering the yellow world line, which represents a particle moving at 3 times the speed of light.
Note that the yellow particle is moving from the left side of the diagram to the right side. You might notice that I left one of the other world lines (dark blue) in there, along with its plane of simultaneity (light blue). This is to show that the yellow particle moves from after the plane of simultaneity to before the plane. Therefore, from the reference frame of the dark blue person, the yellow particle is moving backwards in time. This is true of anything that moves faster than light; there exists a reference frame in which that object is moving backwards in time.
In advanced physics, sometimes you can come up with theories in which there are particles that always move faster than light. These particles are called tachyons. Despite being intriguing little things, tachyons are usually a sign that the theory needs improvement.
The problem with tachyons is that time travel creates numerous paradoxes. What if you go back in time and kill your grandfather? This is called the grandfather paradox. This can actually be a rather compelling argument against tachyons (unless they never interact). However, there are ways around this, as we can see in countless sci-fi stories. Perhaps history is quite resilient, and it is difficult to change anything so as to create paradoxes. Perhaps at the moment of a paradox, the entire universe disappears. Perhaps the entire universe splits into separate possibilities. Perhaps the rules of the universe just don't make any sense.
Although those are all great premises for a work of fiction, they're kind of far fetched as real physical theories if you ask me. But I think there's another way that Special Relativity allows for a sort of time travel. Under special relativity, you can make time go backwards just by accelerating! This is just a natural consequence of the tilting of the plane of simultaneity. Whenever you accelerate, your plane of simultaneity tilts in the direction of your motion. At a certain distance behind you, the plane of simultaneity is tilting faster than time is moving forward. Therefore, whenever you accelerate, any clocks more than a certain distance behind you move backwards.
I did the math, and this distance is equal to c2/a, where a is your acceleration, and c is the speed of light. That means that if you're in a car, and you accelerate from 0 to 60 mi/hr in 12 seconds facing directly away from alpha centauri (the nearest star aside from the sun), then all clocks on alpha centauri will start moving backwards (if I've done my math right). No paradoxes attached, since once you accelerate towards alpha centauri, the clocks will more than make up for their lost time.
That said, it will be impossible to actually observe the clocks moving backwards because of the manner in which the light would reach us.