  The existence of deconfined quark matter in the superdense interior of
neutron stars is a key question that has drawn considerable attention over the
past few decades. Quark matter can comprise an arbitrary fraction of the star,
from 0 for a pure neutron star to 1 for a pure quark star, depending on the
equation of state of matter at high density. From an astrophysical viewpoint,
these two extreme cases are generally expected to manifest different
observational signatures. An intermediate fraction implies a hybrid star, where
the interior consists of mixed or homogeneous phases of quark and nuclear
matter, depending on surface and Coulomb energy costs, as well as other finite
size and screening effects. In this brief review article, we discuss what we
can deduce about quark matter in neutron stars in light of recent exciting
developments in neutron star observations. We state the theoretical ideas
underlying the equation of state of dense quark matter, including color
superconducting quark matter. We also highlight recent advances stemming from
re-examination of an old paradigm for the surface structure of quark stars and
discuss possible evolutionary scenarios from neutron stars to quark stars, with
emphasis on astrophysical observations.
