The highly regarded science weekly New Scientist has in its current issue, available online, an extraordinary article outlining the possibility of testing a form of warp drive within five years.
Allow me to repeat that: WARP DRIVE IN FIVE YEARS!!!!
It involves an extremely obscure branch of physics called Heim theory, worked out by an extremely obscure physicist, Burkhart Heim, in the 1950s. Heim was trying to reconcile the inconsistencies between quantum theory and Einstein's general relativity, a process postulating at least a few more dimensions than the ones we're familiar with, and along the way he came up with a way to, at least theoretically, propel a vessel many times faster than the ordinary speed of light.
Obviously this has never been tested. Very few people know about Heim theory, and hardly anyone understands it, as the mathematics are incredibly difficult. The person most able to promote Heim theory, Heim himself, shunned publicity due to the disfiguring injuries he suffered in an explosives accident towards the end of WWII. The only aspect of Heim theory that has been tested, in computer simulation at a lab in Germany, involves predicting the mass of elementary particles. In this, Heim theory succeeds wildly beyond the capabilities of conventional physics.
There is at least one advanced research facility that has said, if the math could be clarified for them and if they agreed it made sense, they might be able to physically test it within five years. Even if the test succeeds, of course, that's a long, long way from building the Enterprise-A. But to know it was even possible would completely revolutionize our view of the universe and our place in it. Think of it: Mars in three hours, a star 11 light years away in 80 days.
Boy I wish James Doohan had lived to hear of this. The thought that Scotty's beloved engine room might someday exist would have made his heart soar.