It is not only possible, it has been, technically, proven. Astronauts that have been in space have technically traveled milliseconds into the future. Gravity affects time. The reason for this is unknown, as far as I know. In this way , time travel to the future has been proven. Travel to the past is different. Logically, it must be possible if travel to the future is, but that might not be the case because of paradoxes. However, if you add the multi-world theory in, a theory I don't fully believe in, paradoxes might be self fixing.
Time travel already is possible. You may not realize it, but you're doing it right now. The only problems are that we can't travel back in time, and traveling into the future isn't fast enough just yet. Currently we can only travel through time at a rate of 1 second per second.
We may not be able to travel into the past, but we can at least look into the past. Light from the sun takes about 8 minutes to reach Earth. This means that the sunlight you see is really light from 8 minutes ago.
I fully believe that time travel IS possible. All of the mechanics are known about building one so it is fully well possible. Though, because changing the past can have ripple effects in the future, it should only be available to the trustworthy only. But, I hope that one day, it really can be possible,
Think about it, if time travel is possible, then where are the tourists from the future? For time travel to exist, that would mean every seconds of our universe's existence would have to happen at the exact same time for us to travel to it. This means that there would have to be and infinite number of other universes. But we don't understand time enough to manipulate it. Time isn't something humans made up in our brains, it's a field in space that has yet to be understood, it is so complex I doubt we ever will fully comprehend what time actually is.
Time is a constant and natural process; a one way process. Although we are effectively travelling through time into the future, the question is whether we could go into the distant past or distant future. The issue is how one could go back to a moment in time where everything would be as it was at that point. Would the process of time travel reverse what has happened, meaning that everything that had previously existed or occurred after that point is erased, with the outcome of what will come depend on the new present - that alone raises many questions and paradoxes. The grandfather theory is a perfect example of why it couldn't be possible.
So many people seem to belive that it's possible due to the horrible theory of relativity, which has no evidence. (But clocks run faster in space D:) yes they do, and there are other explanations for this. But everyone in the science community have their heads up the butt of Albert Einstein, so they go along with his assumption that light is constant and state it as a fact.
Though clocks can go out of time the more/less resistance you put on them, if that's not more exciting than time travel I don't know what is?
Time seams to exist as a single line. That there is this timeline and no other. If other timelines do or can exist, they would be invisible to us.
Hypothetical 1(if it was possible): You have a time machine and you want to kill Hitler before his rise to power. You send an assassin an nothing changes. Why? Assuming he did travel back in time, a new branch of the timeline would exist. Assuming he was successful, he would only kill him in the new timeline and not the one you exist on. If he was not, this would lead to a paradox. If Hitler never rose to power, you would have never heard of him so would not have sent the assassin. Seeing that you never sent the assassin, he would still exist. This means that the only way a change could be made would be for the timeline to split at the moment the person arrived back in time. Seeing that no difference could be seen, it would be assumed that the time machine did not work and just vaporized whoever when in.
Going back in time would also pose a serious risk, not just to the person but to all mankind.
Hypothetical 2: You have a time machine and travel back exactly one year. The problem is, one year from the time you arrive you will travel back to that same moment. This means that your mass would instantly double. Though this would probably kill you alone, the trouble does not end there. The you that just arrived also would have a you that sent him back so instead of double, your mass would triple. Yet again, this is not the end. Because this would happen an infinite number of times, you mass would increase to being infinite. Basically you instantly become a massive black hole destroying yourself as well as the new timeline that would be created if not both timelines.