  We review the equation of state of matter in neutron stars from the solid
crust through the liquid nuclear matter interior to the quark regime at higher
densities. We focus in detail on the question of how quark matter appears in
neutron stars, and how it affects the equation of state. After discussing the
crust and liquid nuclear matter in the core we briefly review aspects of
microscopic quark physics relevant to neutron stars, and quark models of dense
matter based on the Nambu--Jona-Lasinio framework, in which gluonic processes
are replaced by effective quark interactions. We turn then to describing
equations of state useful for interpretation of both electromagnetic and
gravitational observations, reviewing the emerging picture of hadron-quark
continuity in which hadronic matter turns relatively smoothly, with at most
only a weak first order transition, into quark matter with increasing density.
We review construction of unified equations of state that interpolate between
the reasonably well understood nuclear matter regime at low densities and the
quark matter regime at higher densities. The utility of such interpolations is
driven by the present inability to calculate the dense matter equation of state
in QCD from first principles. As we review, the parameters of effective quark
models -- which have direct relevance to the more general structure of the QCD
phase diagram of dense and hot matter -- are constrained by neutron star mass
and radii measurements, in particular favoring large repulsive density-density
and attractive diquark pairing interactions. We describe the structure of
neutron stars constructed from the unified equations of states with crossover.
Lastly we present the current equations of state -- called "QHC18" for
quark-hadron crossover -- in a parametrized form practical for neutron star
modeling.
