Travelling into the future is completely possible due to the effects of special relativity. If you move through space very fast, time will slow down for you compared to the rest of the Universe. You will only have travelled for a few minutes, but everyone else might have aged by several decades. So travelling to the future is definitely a real thing and astronauts do it all the time (on a small scale). As for travelling to the past, then it gets a little bit fuzzy.
The best answer is really to say nobody knows yet. There are some experiments in quantum mechanics such as the delayed choice quantum eraser experiment which do give the impression (possible illusion) of events in the present affecting choices in the past but it's very hard to confirm that this really is what's happening. In quantum field theory the notion of CPT conversation shows that time-reversal is allowed mathematically, and in fact may be happening at the particle level all the time, but this is again only an interpretation of experimental data and what's going on may be something else entirely.