Phases of Dense Matter in Neutron Stars

After a brief history of neutron stars and supernovae recent developments are discussed. Based on modern nucleon-nucleon potentials more reliable equations of state for dense nuclear matter have been constructed. Furthermore, phase transitions such as pion, kaon and hyperon condensation, superfluidity and quark matter can occur in cores of neutron stars. Specifically, the nuclear to quark matter phase transition and its mixed phases with intriguing structures is treated. Rotating neutron stars with and without phase transitions are discussed and compared to observed masses, radii and glitches. The observations of possible heavy $\sim 2M_\odot$ neutron stars in X-ray binaries and QPO's require relatively stiff equation of states and restricts strong phase transitions to occur at very high nuclear densities only.