Einstein's theory of relativity does not rule out the possibility of time travel, it actually allows for it. And we know he was right, the atom bomb was a result of his equations. M theory allows for a multiverse, or parallel universes, that would rectify the time travel paradox involving affecting the past and altering your present.
The ability to travel into the future is a scientific reality. Russian cosmonaut Sergei Avdeyev holds the record for the most time spent in space, and he also holds the record for time travel. He has spent 749 days in space, traveling at 16,000 mph, which has sent him into the future a fraction of a second. The rate of the passage of time is affected by velocity, we know this, and we actually use this knowledge in our everyday life. A satellite in orbit moving at 20,000 mph loses 2/100ths of a second per year, and we have programmed their computers to compensate for this time differential, otherwise their clocks would continue to fall farther and farther behind compared to ours here on Earth. This method of time travel into the future involves the speed of light and its being constant for every observer no matter their velocity or location, and is based on science fact.
Another theory involving travel into the future involves the warping of the spacetime fabric of our universe by gravity wells or black holes. The warping effect of the black hole on space time alters the passage of time. Relativity predicts that at the centre of a black hole is an infinitely dense point, called a singularity, within which all the normal laws of physics no longer apply. If you were to approach the event horizon of a black hole, your perception of time would not change, but an observer outside the event horizon would observe a change in your passage of time. If you were then to leave the area of the event horizon, little time would have passed for you, but hundreds of years could have passed for the observer outside the area warped by the black hole. Also, Einstein's equations show that such a singularity doesn't just make a dip in the imaginary rubber sheet of spacetime; it makes a tunnel that goes right through and momentarily opens out on the other side. Where is the other side? No one knows, it could be anywhere, in the past, the future, even a different universe. This method of time travel involves theory and observations, but has not been tested.
Traveling into the past is more difficult, but still a scientific possibility. Here we get into the concept of wormholes and the "Grandfather" paradox. Einstein's equations describe a smooth fabric of space time, but his equations are useless at the quantum scale. They can't be used to describe the behavior of sub-atomic particles, what happens at the singularity of a black hole or at the moment before the big expansion (Big Bang).
Along comes quantum mechanics to describe the sub-atomic. Then we see that there seems to be evidence of parallel universes. The particles that we see exist in our universe, but their behavior seems erratic, unless they are interacting with particles in another unseen universe. These observations as well as Heisenberg's Uncertainty principle (basically you can not measure a particles directional momentum and it's position at the same time) gives evidence for a multiverse, in which a time traveler could do whatever they wanted, but would not be able to alter the history of the universe that they came from, instead they would give rise to an alternate reality. The only testable way of proving all this though is to build a time machine and travel back in time.
So, what possibilities are there for traveling into the past? There are three current popular possibilities, two involve light, and the other involves enlarging rips in spacetime at the subatomic scale.
The first possibility is the one depicted in movies such as Star Trek and superman, and involves moving faster than light, which if it were possible, you could leave and get back before you left. It also involves black holes. A phenomenon known as frame dragging describes how a spinning black hole is able to move empty space. It is the same basic concept that would allow the use of a black hole to travel into the future. In the rotating region of space around the black hole, one could travel at speeds up to the speed of light, but from an observer's point of view outside the black hole, you would appear to be moving faster than the speed of light. And for the observer, you would disappear from sight as you travel back into the past.
The second possibility is being developed right now by a man named Ron Mallett, and is based on the effects of a black hole on spacetime from the first possibility. It starts with very powerful rings of laser light that twist time into a loop. Einstein predicted, but was never able to prove this effect of light. With a circulating beam of light it is possible to create a rotating region of space which according to Einstein would also twist time, because space and time are linked. So if we think of our time line as a line from the past to the present to the future, by twisting time and creating a loop, one can travel from the past, to the present, to the future, but because it is a loop, one can go from the future to the past. Of course this means that you could travel back in time to a certain point, you could not travel past the time at which the machine was turned on.
The third possibility involves wormholes. At the quantum levels tiny rips in the spacetime fabric occur regularly. If these rips can be held open and expanded, it may be possible to travel to any point in time or space. To be honest, I have not done much research on this particular possibility, and therefore I do not feel qualified to explain it further.