  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.
