Alcubierre's "Warp Drive"
..Violations of this law lead to causal paradoxes, in which actors in the present can alter their own past. Yet Alcubierre shows that his warp drive does not, in fact, lead to such violations. The reason is that light also travels in spacetime, and is carried along just as the spaceship is. The light beam is still traveling at the speed of light, relative to the spaceship, which itself is not accelerating relative to the spacetime in its immediate vicinity. Although Alcubierre's warp drive does not engender any causal paradoxes, one might still be concerned for a space traveler's welfare. To get to a distant star and back in only moments, the traveler would have to be accelerated at a very large rate, effectively turning him into soup. Or at least this would be true, if not for the fact that accelerations are relative in general relativity.
Recent updates with reduced use of negative energy:
A graph of the curvature of space (or "warp bubble"):
NASA has a bit of info on it:
Search result of "warp drive" from 1994-2002: