  Nuclear matter and finite nuclei exhibit the property of superfluidity by
forming Cooper pairs. We review the microscopic theories and methods that are
being employed to understand the basic properties of superfluid nuclear
systems, with emphasis on the spatially extended matter encountered in neutron
stars, supernova envelopes, and nuclear collisions. Our survey of quantum
many-body methods includes techniques that employ Green functions, correlated
basis functions, and Monte Carlo sampling of quantum states. With respect to
empirical realizations of nucleonic and hadronic superfluids, this review is
focused on progress that has been made toward quantitative understanding of
their properties at the level of microscopic theories of pairing, with emphasis
on the condensates that exist under conditions prevailing in neutron-star
interiors. These include singlet $S$-wave pairing of neutrons in the inner
crust, and, in the quantum fluid interior, singlet-$S$ proton pairing and
triplet coupled $P$-$F$-wave neutron pairing. Additionally, calculations of
weak-interaction rates in neutron-star superfluids within the Green function
formalism are examined in detail. We close with a discussion of quantum vortex
states in nuclear systems and their dynamics in neutron-star superfluid
interiors.
