Is time travel possible? The answer depends on what exactly you are referring to by 'time travel." It appears at first sight to be a simple phenomenon, going backwards and forwards in time, as if time were an elevator and we pushed the buttons to reach different floors. But this is an oversimplification based upon the popular view of time travel as seen in television shows like Doctor Who.
In the Whoniverse time travel is possible. But we do not actually see it. We see somebody enter a police box, then exit it in another place or time. To visualize the process of time travel we are shown the 'howlaround' graphics used in the first opening sequence in the first visualization, and in later ones we will usually see the TARDIS travel through the 'time vortex', a tunnel of glowing light of various intermingling colors.
But this is all fantasy and as much as Doctor Who is 'science fiction', it is set in another type of science to our universe. In our universe time does not exist. Time is merely the observed relation between objects we see. If, for instance, a car passes by us, we remember that it went from point A to point B and we were in the middle, at point C. Thus it went past us and this took some time.
However, this is our memory playing a trick on us. There is no time, just change. The car went from point A to point B alright, but this was the universe changing. The car, part of the universe, moved its position. But time is only existent in our memory as the difference between states of the universe around us.
Time, often referred to as the fourth dimension, does exist mathematically however. Using time we can predict events by using patterns such as seconds and hours, but this is all relative to our position and context. There is no universal 'time', because different parts of the universe travel at different speeds. Gravity is one such example of a difference between the speed of an object in one position and in another, and since time is the measurement of speed, time changes with gravity.
If we imagine a theoretical universe with a state of inertia and a beach ball being the only component of the universe, there would be no such thing as time. The beach ball, unaffected by any force and containing no energy, would remain static and not move. Our perspective of the ball would thus be the only change, and that would require us to exist within that universe. Thus time is the observed relation between objects, speed.
Speed is a complex thing to explain. It is all relative. Imagine two scenarios, in the first a baseball is dropped at 100 miles per hour from the top of a tower 500 meters high. The time taken by the baseball is an amount (imagine there is no wind resistance or other affects besides gravity, and that the ball was dropped without any force downwards given by the dropper). In the second scenario the exact same process happens, but in this scenario the universe, and all matter contained therein, is a billion times larger in size. The relativity remains the same, and if you traveled to the first and second universe and viewed both events you would think you had seen the same event twice.
So which ball fell the fastest? In the smaller universe there is less distance for the ball to travel, but it goes slower, and in the larger there is more distance and the ball goes faster. Thus they would be going at the same speed, because speed is measured relative to the context in which the measurement takes place.
Time does not, therefore, exist as a substance, entity or dimension beyond theoretical mathematics, there is only the changing state of the universe. Imagine once more the last scenario with the baseball drop. The baseball started at the top of the tower and ended at the ground. But at any point in time it was only in one position. If you were now on the ground with the ball, there would be only the present, and as such the ball never dropped in the first place. There is only the present. The past does not exist, and neither does the future. There is but a single 'frame' of time, which changes.
As such the only way to 'time travel' would be to gain control of the entire universe and reverse its forces so that every process goes backwards, interacting in an equal and opposite fashion to how it originally operated. Newton's third law of action and reaction says as much. The only way to go forwards in time would be to speed up the entire universes forces. Both operations, of course, would require immense power, and it is not likely we will ever attain such power.
So is time travel possible? Not unless such power was gained. Theoretically is it possible? Perhaps in another context where Newton's laws are not obeyed and time refers not to relative observations, but to some 'fourth dimension' that actually exists. I have tried for many years to find this elusive fourth dimension but to no avail. Mathematically it can exist, however, but physically only three dimensions appear possible in any imaginable scenario.
Thus unless there is a theoretically possible fourth dimension that can actually exist beyond mathematics into reality, time travel is impossible, because even if you did gain control of the entire universe and reverse it, it would, in what we call 'time', i.e the observed change of the universe recorded as a series of events, take place after the original past events had taken place.
Technically you could with this theoretical super power revert the universe to, say, the 1970s. People would look exactly as they did in the 1970s. Technology would be just as advanced as in the 1970s. People would be doing the exact same things they did in the 1970s without a single change (unless quantum physics is as random as it appears to be in which case there might be minor changes but nothing major), but without a doubt it would not be the same 1970s as existed before the universe was reverted.