Detections of gravitational waves from binary neutron star mergers offer an exciting, new approach to constraining the equation of state (EOS) of ultra-dense matter. In this talk, I will summarize what we have learned about the EOS from the first binary neutron star mergers and what we might hope to learn from future events. I will start by discussing a new one-to-one mapping between the tidal deformability measured from a neutron star merger and the stellar radius, and I will compare the radius that was inferred from GW170817 to previous X-ray measurements of neutron star radii. I will also present new constraints on the nuclear symmetry energy from GW170817, which point to smaller values than are typically found in laboratory-based nuclear experiments, and which reinforce the radius constraints. In the final part of the talk, I will introduce a framework for extending models of the cold EOS to arbitrary temperatures and compositions, which will enable us to disentangle the role of thermal effects from the underlying cold EOS in future analyses of gravitational wave events. I will finish with some preliminary results from a set of merger simulations that explore the parameter space of this new framework.