Einstein in his theories of spacetime used reference frames that were in motion relative to each other. The special theory of relativity deals with the mechanics of particles and the electromagnetic field as measured by two non-accelerating inertial observers in uniform motion relative to each other. The surprising result was that moving clocks tick slower, moving rods contract in their direction of motion, and mass is equivalent to energy. All of these results follow from the principle of relativity plus the idea that the speed of light in vacuum is "invariant" for these non-accelerating observers. These predictions have been checked by many many experiments and they cannot be seriously disputed.
However, it is also sometimes claimed that the speed of light is an absolute barrier. That result is not fundamental to special relativity without an additional postulate called "the principle of retarded causality". This principle asserts that effects are always after their causes in all frames of reference. In fact, this principle violates quantum mechanics. This apparent clash between quantum mechanics and causality is the most important problem in physics today IMHO.
Bell's theorem shows that the statistical predictions of quantum mechanics require either that the universe consists of parallel worlds, or that there are faster-than-light influences at the individual quantum event level, or both. Experiments confirm quantum mechanics and violate a locality inequality derived by the late John Bell. The experimental loopholes are not significant in the judgement of the experts. Eberhard's theorem shows that while the statistical predictions of quantum mechanics violate retarded causality at the individual event level, these violations cannot be used to send and receive messages faster-than-light and backward-in-time because of uncontrollable random quantum fluctuations that cover up the anomalous effect.
Beyond these important results, a very controversial attempt is now underway to create a really new physics of consciousness. The idea is that the mind is a kind of giant quantum wave function of an intelligent network of nonlocally correlated quantum switches in the brain called "Eccles Gates". These gates are single electrons that control protein shapes in microtubules of nerve cells. There is a new feature of feedback-loop control that goes beyond standard quantum mechanics allowing the violation of the statistical predictions of quantum mechanics in such a way that causality-violating messages are sent and received by different parts of the brain both faster-than-light and even backward-in-time.