  Updated constraints from neutron star masses and radii impose stronger
restrictions on the equation of state for baryonic matter at high densities and
low temperatures. The existence of two-solar-mass neutron stars rules out many
soft equations of state with prominent "exotic" compositions. The present work
reviews the conditions required for the pressure as a function of baryon
density in order to satisfy these new constraints. Several scenarios for
sufficiently stiff equations of state are evaluated. The common starting point
is a realistic description of both nuclear and neutron matter based on a chiral
effective field theory approach to the nuclear many-body problem. Possible
forms of hybrid matter featuring a quark core in the center of the star are
discussed using a three-flavor Polyakov--Nambu--Jona-Lasinio (PNJL) model. It
is found that a conventional equation of state based on nuclear chiral dynamics
meets the astrophysical constraints. Hybrid matter generally turns out to be
too soft unless additional strongly repulsive correlations, e.g. through vector
current interactions between quarks, are introduced. The extent to which
strangeness can accumulate in the equation of state is also discussed.
