Multimessenger observations of compact stellar remnants offer a unique opportunity to study the behavior of matter in the most extreme environments anywhere in the universe. I will review what recent gravitational-wave, x-ray, and radio observations of neutron stars, along with nonparametric hierarchical Bayesian inference, tell us about the nuclear equation of state at densities several times larger than what is accessible through terrestrial experiments. We will discuss implications for both neutron stars’ properties and modern theories of nuclear interactions. I will then introduce several applications that rely on knowledge of the equation of state and show how gravitational waves from compact binary coalescences can be used to study other astrophysical phenomena.