Time travel may be possible, but to understand how, we should begin by assessing the relativistic physics relevant to the concept of time travel. Relativity tells us that time moves more slowly for objects moving at high speed. For objects moving at the speed of light, time stops. As such, if you were to take a lap from the earth to the sun and back at the speed of light with a stopwatch, you would perceive that journey as instantaneous and your stopwatch would read 0 minutes and 0 seconds. But from my perspective, the journey would take you 16 minutes, and when you returned, my stopwatch would read 16 minutes. Effectively, you traveled 16 minutes forward in time.

To travel backward in time, we need to think about graphing the "speed" of time as a function of the speed of the object experiencing it. We can draw x-y axes and a simple line with a negative slope, intersecting the x axis where the object's speed equals the speed of light. Time would then have negative speed at object speeds faster than the speed of light. Thus as you say, to travel back in time, you would have to travel faster than the speed of light. So far, no physical object is known to be able to do that. The fastest objects measured on earth have been subatomic particles in the large hadron collider at CERN in Switzerland, at 99.99% the speed of light.

The mathematical framework of string theory, proposed to unify the observations of quantum mechanics with Einstein's relativistic physics, predicts 10 spatial dimensions to exist. If these dimensions do exist (and as of now this is very hard to prove or disprove), a type of time travel could be possible by means of higher dimensional transport. The key concept is explained in the movie (watch the scene at Interstellar ), where the piece of paper is a 2-dimensional world, and travel within those two dimensions has to take place in a path that can be drawn across the page.

In our higher-dimensional 3-D world, we are free to fold the paper out of plane and travel between two points on it without occupying the 2-D space between them. If we extend our thinking to a spatial dimension beyond our 3-D spatial world, "folding" through a higher dimension might enable us to arrive at a different point in space. In fact, we already do that: think of a 3-D spatial world without time, meaning the whole 3-D world is just a snapshot of one moment in time. If we want to travel to another point in that 3-D world, we actually have to spend time doing that. In that sense, time is our 4th dimension, that allows us to come to a different spot in 3-D space. Thus to travel in 2-D space without time, you need 3-D space, and to travel in 3-D space, you cannot travel without time. By extension, to travel arbitrarily in time, you must use a higher dimension. Spatial dimensions higher than 3 are hard for us as humans to perceive, which is one of the reasons we have not made much progress on time travel to date.

According to the special relativity, if an object's initial velocity is lower than the speed of light, it is not possible to accelerate it over the speed of light. Calculation shows that it would cost infinite energy to accelerate this object to speed of light. This spares us from worrying about all the paradoxes (such as time-travel and all the sci-fi related paradoxes) if things can travel above the speed of light.

So far as we know in the practical sense, time travel is impossible, and so is faster-than-light (FTL) travel. In fact, under our current theories, time travel and FTL travel would be the same thing. We have never observed anything traveling backwards or forwards in time, nor have we ever observed anything traveling faster than light.

There is a theoretical construct called a Tipler cylinder, which is a rotating massive cylinder that can twist space-time as it rotates, in such a way that it could cause time travel. However, as I understand it, such a cylinder would have to either be infinitely long, or involve negative energy. There are various other theoretical negative energy constructs that could do similar things, including FTL travel. Negative energy, however, is a bit of a question mark: the expansion of the universe may be driven by it, but if so, there is no way for us to control it.

Of course, all of this depends on our current theories of physics, which have a lot of holes with regard to the mechanism of the expansion of the universe. As such, it is conceivable that FTL travel may yet be possible. However, the Fermi Paradox (the lack of evidence for extraterrestrial intelligence, despite our own existence) does suggest that it is not.

Time travel isn't possible in the science fiction sense, but this link is cool:
here.

“There is no limit to how fast the universe can expand, says physicist Charles Bennett of Johns Hopkins University. Einstein's theory that nothing can travel faster than the speed of light in a vacuum still holds true, because space itself is stretching, and space is nothing.Sep 10, 2013”. “May 2, 2016 - We are told that nothing can travel faster than light. This is ... Light is faster than anything else”.