Various scientists, like Stephen Hawking, Kip S. Thorne, Michio Kaku, etc, have already developed some gedanken experiments that illustrate the reality of time travel. We just do not have the means to accomplish this yet.
For instance, you could maybe consider the twins paradox as a form of time travel. You have two twins, and one travels on a ship capable of light-speed while the other stays on Earth. Also, assume that the twins are 20 years old, and the travelling twin makes a round-trip journey of, say, 80 light-years. Due to the relativistic effects of light-speed travel, the travelling twin will come back being 20 years old - the exact same age as he left - while the other twin, as well as the rest of the universe, would be 80 years older. Because of this, one could think of the travelling twin as having jumped forward in time.
Similarly, bodies with some mass will also warp space-time to the point that time slows down, even a tiny up or down quark. But the effect is much more pronounced around an extremely massive body, like a black hole, that can have masses up to billions of our suns. If you enter the immediate vicinity of such a black hole, you will experience time at the normal rate, but when you exit that region, you may find that many years have actually passed, even though you spent only an hour in the gravity well.
There is a connection between mass and light-speed. For instance, the closer to light-speed you travel, the more mass you will actually have. As you approach light-speed, your mass will increase to infinity, which will ultimately slow down time around you, which is the root cause of the twins paradox. (This is also the reason why faster-than-light-speed travel is not possible: Imagine the force required to accelerate an infinite mass! The inertia would also be infinite. In other words, accelerating an infinite mass is not possible.)
Since we do not have light-speed travel, nor do we have easy access to a black hole (fortunately), we get that the main problem with time travel is that manipulating the space-time continuum would require a considerable amount of energy in electron volts to accomplish.... In fact, it would require more than we can ever produce on planet Earth with our current technology. Besides, if we did have that technology, would it really be wise to use it? I believe that the most important field of research right now does not lie along those lines.