  The equation of state (EoS) of hot and dense matter is a fundamental input to
describe static and dynamical properties of neutron stars, core-collapse
supernovae and binary compact-star mergers. We review the current status of the
EoS for compact objects, that have been studied with both ab-initio many-body
approaches and phenomenological models. We limit ourselves to the description
of EoSs with purely nucleonic degrees of freedom, disregarding the appearance
of strange baryonic matter and/or quark matter. We compare the theoretical
predictions with different data coming from both nuclear physics experiments
and astrophysical observations. Combining the complementary information thus
obtained greatly enriches our insights into the dense nuclear matter
properties. Current challenges in the description of the EoS are also
discussed, mainly focusing on the model dependence of the constraints extracted
from either experimental or observational data (specifically, concerning the
symmetry energy), the lack of a consistent and rigorous many-body treatment at
zero and finite temperature of the matter encountered in compact stars (e.g.
problem of cluster formation and extension of the EoS to very high
temperatures), the role of nucleonic three-body forces, and the dependence of
the direct URCA processes on the EoS.
