In real space, even approaching lightspeed would require more energy per second than all of Earth's industries use per year. While this may be explained by advances in technology, the general formulas for velocity and acceleration are such that as you approach the speed of light, the energy needed to accelerate anything with non-zero mass increases asymptotically. In other words, you need an infinite amount of energy (and an infinite amount of time) to accelerate to the speed of light. And expelling infinite amount of propellant - unless you have Reactionless Drive, which is a huge can of worms all in itself.
Additionally, Albert Einstein's well-supported Special Relativity theory holds that FTL travel simply cannot work, by the fundamental laws of physics. Due to the space-time distortion at relativistic speeds, it would imply violations of causality itself! For example, a superluminal signal sent from A to B will appear, in another reference frame, to be received before being sent. This is the reason why some say that FTL travel implies time travel - but that is sloppy language. What actually happens is that causality is thrown out of whack, and that different observers don't always agree whether effects precede their causes or not. In effect, this means that, in special relativity, going faster than light is impossible - with or without time travel. In Einstein's much more comprehensive theory of General Relativity, however, matters are considerably more complex, and FTL travel is not known to necessarily imply causality violations (see Real Life examples)...
But, all in all, suffice it to say that having the protagonists' rocketship outrun a beam of light will always require some serious hand waving by the author. See also Time Dilation.
Even outside of Einstein's flavor of Relativity, other theories tend to impose limitations just as strict or more, because they all are intended to describe the same effects of matter moving at significant portion of lightspeed in the first place. Thus what it may take to make a "warp drive" believable is to remove "gravity as space curvature" concept and then add another Physics extension that brings space curvature back in some other way.
Of course, any space curvature intense enough to have such use by definition must have much the same "little side effects", like refraction and tendency to cause deformations trying to rip apart most things passing through it if they are more complex than a simple gas molecule. Note that once the gravity is not the same as spatial anomalies, it's most likely to be affected by them - which may be bad news for massive objects close to even-more-massive objects. Then again, usually it's good news for the authors seeking an excuse to have "No Warping" Zones around anything potentially important.
- It should be noted that in relativistic physics it is possible to get arbitrarily far within an arbitrarily short time from the perspective of the person doing the moving. Time itself is relative, and stationary observers see the flow of time for the person traveling slow down, so that in the time it takes the traveler to get to their destination, he or she experiences less time passing than the stationary observers do. The person making the journey sees the distance to their destination contracted along their direction of travel, so they have less distance to cover than they did when they were stationary. This makes it theoretically possible for an astronaut to cross great distances in a single life-time, but FTL travel is still required if you don't want the journey to take eons from the perspective of those the traveler left behind on Earth.
- Not only because it got Real Life physics issues of its own. It's also a gravity theory and is already implicitly thrown away once the 'verse in question has "gravitons" or "antigravity".
- Those without Riemannian space and "gravity is effect of space curvature - it's the same as inertia" axiom that needs it are likely to eliminate even the loophole of "Alcubierre drive", such as it is.