You may have read the recent news about neutrino speeds being measured as faster than c, the speed of light in a vacuum. They were only measured as about 0.002% faster, but given the presumptive status of c as a hard limit, that's still major news, if true.
The current reaction is cautious; many people seem to expect that the result will turn out to be experimental error of some sort, and that is indeed quite possible. However, the idea of neutrinos travelling faster than light is not unprecedented - and would not necessarily be the problem for relativity theory that news reports on this issue are describing.
Way back in the 1960s, it was noted that while relativity theory implies that normal matter cannot accelerate past the speed of light, the theory allows for faster than light particles - dubbed "tachyons" - that could not decelerate below the speed of light. These particles would have various unusual properties, such as imaginary rest mass. Some were uncomfortable with the concept, as they didn't like the implications with respect to causality, but it didn't contravene any known physical laws.
Decades later - but still starting more than a decade ago now - indirect measurements of neutrino mass resulted in best estimates of the square of the neutrino rest mass being negative, implying an imaginary rest mass. The high end of the 95% confidence interval was positive, so the measurements were interpreted as indicating a zero or small rest mass, but that may need to be reexamined in light of the new information.
In addition, theoretical papers have been published suggesting reasonable physical interpretations of tachyonic particles. For example, the neutrino speed measurement involved neutrinos travelling from the CERN laboratory in Switzerland to the OPERA detector in Italy. In some frames of reference, the neutrino detection by OPERA would precede the emission of those neutrinos at CERN - thus leading to causality questions - but in those frames of reference, the same physical events could be interpreted as emission of antineutrinos at OPERA which were absorbed at CERN, potentially resolving the causality issues. Of course, if one doesn't believe in causality at the level of subatomic physics, as I don't, those issues needn't be resolved in the first place.
Again, all of this may be coincidence, and the speed measurement may simply be the result of a mistake or some unusual physical effect that the researchers hadn't thought of. However, the measurements may also be perfectly legitimate - in which case, while it won't be a revolution in physics, there will be major implications for future technological possibilities.
Neutrinos measured at speeds faster than c:
One measurement of neutrino mass squared with negative best estimate:
Resolution of any causation paradox:
Edit: previous measurement of neutrino speed, nonsignificantly faster than c: