We discuss the physics of matter that is relevant to the structure of compact stars. This includes nuclear, neutron star matter and quark matter and phase transitions between them. Many aspects of neutron star structure and its dependance on a number of physical assumptions about nuclear matter properties and hyperon couplings are investigated. We also discuss the prospects for obtaining constraints on the equation of state from astrophysical sources. Neuron star masses although few are known at present, provide a very direct constraint in as much as the connection to the equation of state involves only the assumption that Einstein's general of theory of relativity is correct at the macroscopic scale. Supernovae simulations involve such a plethora of physical processes including those involved in the evolution of the precollapse configuration, not all of them known or understood, that they provide no constraint at the present time. Indeed the prompt explosion, from which a constraint had been thought to follow, is now believed not to be mechanism by which most, if any stars, explode. In any case the nuclear equation of state is but one of a multitude on uncertain factors, and possibly one of the least important. The rapid rotation of pulsars is also discussed. It is shown that for periods below a certain limit it becomes increasingly difficult to reconcile them with neutron stars. Strange stars are possible if strange matter is the absolute ground state. We discuss such stars and their compatibility with observation. 112 refs., 37 figs., 6 tabs.