  Now 50 years since the existence of the neutron star crust was proposed, we
review the current understanding of the nuclear physics of the outer layers of
accreting neutron stars. Nuclei produced during nuclear burning replace the
nascent composition of the neutron star ocean and crust. Non-equilibrium
nuclear reactions driven by compression alter the outer thermal structure and
chemical composition, leaving observable imprints on astronomical phenomena. As
observations of bursting neutron stars and cooling neutron stars have
increased, the recent volume of astronomical data allows new insights into the
microphysics of the neutron star interior and the possibility to test nuclear
physics input in model calculations. Despite numerous advances in our
understanding of neutron star interiors and observed neutron star phenomena,
many challenges remain in the astrophysics theory of accreting neutron stars,
the nuclear theory of neutron-rich nuclei, and the reach and precision of
terrestrial nuclear physics experiments.
