This place seems to be a "tough" room and most of the people who post here appear to be skeptics.
Here are some personal thoughts on time travel. Some may seem rather obvious provide a starting point.
First we do definitely travel through time. Time seems to pass us by at a definite rate. Something happens even if we appear to stand still. As the ancient Romans said tempus fugit (time flees).
So we are always traveling to the future. What we usually mean by time travel is the question of whether or not we can increase the rapidity with which we travel through time or even the direct as what would happen if we could travel to the past. Almost everyone has seen or read The Time Machine by H. G. Wells, an early sci-fi writer, in which we are taken on a tour of the future. Einstein's Theory of Relativity answered the question about the rapidity of travel in the affirmative. Time dilation (lengthing or stretching) says that a person traveling at speeds approaching that of light will experience time at a slower rate. If both a stationary observer and a moving observer synchronized their clocks, and t was the time between two events as measured by the stationary observer, then that of the moving observer is t' = t* sqrt(1-v^2/c^2). The literature mentions the "twin paradox" which discribes what would happen if one twin takes a space trip near the speed of light and returns to Earth. The theory says that the traveler will find that his has aged at a much higher rate than he has. So one might try to travel to the future this way.
The case for travel to the past is less optimistic. We can communicate with the future but communication with the past is unknown of. There are some who claim to have this power no scientific proof that I know of. This is called clairvoyance and I think that the probability of this happening is much greater than a person actually traveling back in time. In summary, we appear to be well connected to the future but not at all well connected with the past.
The case is not very clear at the subatomic level. In Quantum Electrodynamics (QED), one allows for communication with the past. Use is made of anti-particles with negative energies. These can be viewed as ordinary traveling backwards in time. So there might be a channel through which we can communicate with the past. If we can communicate with the past then it might be possible to build a machine, a "transporter", which will actually send a person back in time. This is the premise for Timeline as I have already mentioned.
Finally, I would like to mention that the pyramids of the ancient Egyptians can be viewed as a kind of time machine. There is no doubt that this is what they were intended to do. The religion of the time said the the soul (the ba), continued on after a person died and would be reunited with the person's future body (the ka). If one views the pyramids as a communications device, then it could be said that they were successful time travelers. One could live beyond one's natural lifetime and be around for one's descendents. Many cultures today believe in anscestor worship and this provides a kind of continuity. Some of the more successful at this type of timetravel were refered to as gods. They tried to achieve immortality.
There are modern efforts today to do this. This is refered to as cryosuspension. A dead body is frozen in liquid nitrogen in the hope that at some future time it may be revived.
Live cells can be frozen in liquid nitrogen and then later thawed out. At low temperatures, molecular events occur at a much slower rate so the cells can be preserved over long periods of time and revived. This idea was used in 2001 to allow the scientists to go to Jupiter without needing to take as many provisions along. This method has also been suggested as a means for time travel to the future.