The cooling of a compact star depends very sensitively on the state of dense matter at supranuclear densities, which essentially controls the neutrino emission, as well as on the structure of the stellar outer layers which control the photon emission. Open issues concern the hyperon population, the presence of meson condensates, superfluidity and superconductivity, and the transition of confined hadronic matter to quark matter. This paper describes these issues and presents cooling calculations based on a broad collection of equations of state for neutron star matter and strange matter. These results are tested against the body of observed cooling data.

Invited review for a special issue of Nucl. Phys. A. 44 pages, 18 figures