Abstract Using results from energy calculations of neutron matter, we construct various neutron star equations of state. From these equations of state, together with the Tolman-Oppenheimer-Volkoff equations, we calculate quantities such as pressure, mass density, mass energy density, total mass, radius, and moment of inertia for neutron stars. Comparison is made with calculations based on other nuclear potentials and nuclear energy calculations, and our results are in reasonable agreement with results from observational data. We also calculate mass, radius, moment of inertia and surface gravitational red shift of “quark stars” described by three models: a non-interacting Fermi gas model, an asymptotic MIT bag model, and a perturbative QCD model. We also give results of phase transition calculations, and comment on possible observable differences between neutron stars and quark stars. Finally, properties of “hybrid stars” consisting of a core of strange quark matter surrounded by ordinary neutron matter are investigated. We discuss star models based on phenomenological equations of state including a phase transition between the hadronic phase and the quark-gluon plasma. For certain parameters, these equations of state support the existence of hybrid stars. The identification of such objects could provide information on the properties of strange quark matter.