What are some of the current theories on time travel? Have any experiments been done to test these theories?
Thankee 
For the most part, it is considered impossible.
There are some possible minor exceptions if you start talking about black holes, worm holes, universe birth/death, or maybe some other oddities. Understandably these are hard to test.
You can also have enormous ( like solar-system sized ) cylinders spinning at relativistic speeds, and near the edge, space and time dimensions can get switched (per Einstein). Understandably, we don’t have the technology to do that, nor do we expect to anytime soon.
As far as I know, there are no other scientifically reasonable approaches.
Ivan Semeniuk wrote the cover story for the September 20-26 issue of New Scientist. It can be found at http://www.physics.utoronto.ca/~peet/media/NewSciTimeTravel/
The gist of the article is that while general relativity allows time travel as a solution to its equations, in practice the universe “conspires” to block any travel in time. However, there is a small possibility that string theory might show these conclusions to be in error.
Read the article. It’s well worth the time.