Of the theoretical means for achieving faster-than-light travel, the most plausible one is the “warp” drive, where a ship travels at superluminal speeds by creating a bubble of space behind it, while compressing the space in front of it. The ship would not move inside of the bubble, but would be carried along with it, like a wave. The upside of this is that it achieves FTL speeds while avoiding time dileation and other relativistic effects. In other words, you can travel across the galaxy and not worry that thousands of years have elapsed on Earth in your absence.
However, as Jason Major details at io9, there is one important downside to warp travel. The energy built up during the voyage could incinerate everything around it:
Space is not just an empty void between point A and point B… rather, it’s full of particles that have mass (as well as some that do not.) What the research team - led by Brendan McMonigal, Geraint Lewis, and Philip O’Byrne - has found is that these particles can get “swept up” into the warp bubble and focused into regions before and behind the ship, as well as within the warp bubble itself.
When the Alcubierre-driven ship decelerates from superluminal speed, the particles its bubble has gathered are released in energetic outbursts. In the case of forward-facing particles the outburst can be very energetic - enough to destroy anyone at the destination directly in front of the ship.
In Star Trek: Deep Space Nine, ships are constantly coming out of warp near the station and its surrounding planets. If each ship is creating its own set of energy bursts, then there’s no way that the station (or the planets) could survive. For the Star Trek (and Star Wars) universe, this makes galactic travel very difficult, if not impossible.
This is all to say that of the solutions to FTL travel I’ve read or seen, the best comes by way of John Scalzi. In his book Old Man’s War, the Colonial Defense Forces—a military organization responsible for populating and defending humanity’s interstellar colonies—travel through use of the “skip” drive. Here’s how Scalzi describes it:
“All right, look,” Alan said. “You asked me how the skip drive works. And like I said, it’s simple: It takes an object from one universe, like the Modesto, and pops it into another universe. The problem is that we refer to it as a ‘drive.’ It’s not really a drive at all, because acceleration is not a factor; the only factor is location within the multiverse.” […]
"The point is: multiple universes. The multiverse. What the skip drive does is open a door to another one of those universes.”
I have no idea whether the science behind this is real or not, but the idea of FTL travel that works through “creating” new universes and moving objects into them is fascinating.