Regarding our current understanding of physics to be solid and whether FTL is possible, I would point out we are seeing hints of FTL already, at least in massless matter. Theoretical negative matter is interesting as well. One exception to the "rule" throws into doubt the absoluteness of the rule and opens theoretical ways to manipulate it. Theoretically, of course.
Quantum entanglement moves faster than light.
If I have two electrons close together, they can vibrate in unison, according to the quantum theory. If I then separate them, an invisible umbilical cord emerges which connects the two electrons, even though they may be separated by many light years. If I jiggle one electron, the other electron "senses" this vibration instantly, faster than the speed of light. Einstein thought that this therefore disproved the quantum theory, since nothing can go faster than light.
But actually this experiment (the EPR experiment) has been done many times, and each time Einstein was wrong. Information does go faster than light, but Einstein has the last laugh. This is because the information that breaks the light barrier is random, and hence useless. (For example, let's say a friend always wears one red sock and one green sock. You don't know which leg wears which sock. If you suddenly see that one foot has a red sock, then you know instantly, faster than the speed of light, that the other sock is green. But this information is useless. You cannot send Morse code or usable information via red and green socks.)
Negative matter.
The most credible way of sending signals faster than light is via negative matter. You can do this either by:
a) compressing the space in front of your and expanding the space behind you, so that you surf on a tidal wave of warped space. You can calculate that this tidal wave travels faster than light if driven by negative matter (an exotic form of matter which has never been seen.)
b) using a wormhole, which is a portal or shortcut through space-time, like the Looking Glass of Alice.