Understanding the equation of state (EOS) of neutron-rich matter is a central goal of nuclear physics that cuts across a variety of disciplines. Indeed, the limits of nuclear existence, the structure of neutron stars, and the dynamics of core-collapse supernova, all depend critically on the nuclear-matter EOS. In this talk I will concentrate on the EOS of cold, neutron-rich matter with special emphasis on its impact on the structure and composition of neutron stars. In particular, I will discuss the evolution with density of the many exotic phases that are predicted to exist in a neutron star: from the low-density crust to the high-density core.