Time travel is possible from a couple points of view. One is through Einsteins general theory of relativity, and the other is with quantum field theory. Einsteins theory of general relativity is his special theory with gravity taken into account. This theory applies to the ultra big and fast. Quantum field theory on the other hand is simply early-day quantum mechanics with Einsteins special theory of relativity, which applies to the ultra small and fast. These two theories are currently being combined into an ultimate theory of everything, which is altogether in a totally different direction.
In Einsteins theory time travel is possible in both directions. Even before Einstein published his General Theory of Relativity, we knew that it was possible to travel into the future. His Special Theory of Relativity gave us that prediction. It was his General theory of Relativity that gave us the predictions that it might be possible to travel into the past.
In the case of quantum field theory this question has to be taken very carefully. You see because in this field theory, there is no preferred direction to time. Things can and do travel in both directions of time, just as surely they do through space. An electron traveling backward in time is exactly equivalent to a positron traveling forward in time. So if anything is to travel into the past – it might as well travel into the future. This makes perfect sense if you would stop to think about it. My future will be in someone elses past.