The discovery of gravitational waves from the merger of binary black hole systems, and more recently from the merger of binary neutron stars, have given us a wealth of new information about the astrophysics of these extremely energetic events. They also have given us the opportunity to study the fundamental properties of the gravitational waves themselves, and of the physics of gravity in the strong-field, highly dynamical regime where it has never before been explored, permitting unique new tests of the modern theory of gravity: Einstein's General Theory of Relativity. This talk will focus on recent results from the latter subject. We will review what we have learned in the last few years about the fundamental properties of GWs: their speed, polarization, dispersion, Equivalence Principle, and the mass of the graviton. We will review the tests of general relativity that have been performed so far with binary black hole mergers. Time permitting, we will present results on the local Hubble parameter, and recent estimates of the nuclear equation of state at the hyper-nuclear densities of neutron star cores, from the binary neutron star merger observed in August 2017.