In real physics understanding spacetime is important to understand gravity. So, we saw lots of research in the past two decades about that. Time travel was a byproduct of that. A lot of peer reviewed papers (literally hundreds about the subject) can be found on the net. Most of these are about thought experiments and their physics implications. These are hard to read if you are not into math, quantum mechanics, string theory, and so on. Often they just discuss a part of the problem and you have to search for the missing parts in other (sometimes unrelated) papers. Because theories need to be falsified, reading a paper A that mentions a given method can be proven wrong by B published several years later. That's how it goes. Whilst TT may be theoretically possible, most methods are not concerned with actually sending travelers. Of course all of that also takes the fun away.
I think, because gravity is still not well understood these days and because string theory has seen a revival we see a steady stream of peer reviewed papers mentioning TT/spacetime/gravity in one way or another.
Sometimes you'll find help in a totally unrelated subject. Here is an example.
Another field is that of quantum computing. It is in the stage of actual development. In September 2011 it was proven that quantum computers can be turned into a device using the van Neuman architecture. In Februari 2012 IBM said that, after many years of research, they know enough to create such a device and are actually creating one. Currently it needs extreme low temperatures which can only be created in a lab, so you won't find them on your desk next year. However, in April 2012, yet another international team working on the problem for many years, build a prototype of a computer using two qubits that actually worked in 95% of the time. They claim that the diamond based technology can be scaled to room temperature.
According to David Deutsch, a researcher who worked on quantum computing for a long time, quantum computing is (or more general quantum superpositions are) evidence of a many worlds quantum multiverse. He wrote a book about that called "The Beginning of Infinity", which was published in 2011.
The existence of quantum superpositions have been proven by various experiments long ago, and the principle does make quantum computers work. And now technology has been created to apply the theory.
Why is that important to TT? Well, some TT theories require a many worlds quantum multiverse to make sense of TT and to get rid of any paradoxes (should such paradoxes exist). So, by itself quantum computing is unrelated to TT, but indirectly it helps to find the pieces of the puzzle.