One of the Star Trek franchise's most enduring legacies in science fiction is the fictional "warp drive" technology that enables faster-than-light travel. It's not the kind of thing that can be achieved with conventional rockets, but that doesn't mean it's entirely outside the realm of scientific plausibility. In fact, there's a long tradition of physicists writing speculative technical papers suggesting ways in which a warp drive or hyperdrive might come about.
The problem is that Einstein's theory of relativity says that no object with mass can match or exceed the speed of light, since its mass will increase along with its acceleration. Even in giant particle accelerators like the Large Hadron Collider at CERN in Switzerland, tiny subatomic particles can reach 99.9% of the speed of light without ever matching or surpassing it. So physicists decided that the trick is moving spacetime itself. The latest proposal, just this past June, involved creating a bubble of spacetime using dark energy or some other form of unspecified "exotic matter."
"The idea is that you take a chunk of spacetime and move it," says Marc Millis, who used to head up NASA's Breakthrough Propulsion Physics Project. "The vehicle inside that bubble thinks that it's not moving at all. It's the spacetime that's moving." You can hear more about this from Millis himself in this video at Space.com (embedding disabled), complete with selected clips from the many science fiction films over the years that have depicted faster-than-light travel through space. It's truly a case of Hollywood inspiring science through fiction, and science inspiring Hollywood in turn with a new host of ideas for space travel in the future.