  Implications of recently well-measured neutron star masses, particularly near
and above 2 solar masses, for the equation of state (EOS) of neutron star
matter are highlighted. Model-independent upper limits to thermodynamic
properties in neutron stars, which only depend on the neutron star maximum
mass, established from causality considerations are presented. The need for
non-perturbative treatments of quark matter in neutron stars is stressed
through studies of self-bound quark matter stars, and of nucleon-quark hybrid
stars. The extent to which several well-measured masses and radii of individual
neutron stars can establish a model-independent EOS through an inversion of the
stellar structure equations is briefly discussed.
