Well, since 1905 and Einstein's theories, we have a rule that nothing can exceed the speed of light in vacuum
Since then, no observations have been able to demonstrate phenomenon that truly are exceeding that limit.
It's a bit hard to exactly understant all that implies, but basically it would be grounds for new theories going beyond what we currently use.
Some think we could break the causality principles, exploring time travel... at least for information.
Interresting fact is that we just had Steins Gate talking time travel and CERN