After a general introduction to neutron stars, I will discuss the special role of their inner crust. This region is characterized by the coexistence of a lattice of
neutron-rich nuclei ("clusters") and a gas of unbound neutrons. The unbound neutrons are supposed to be superfluid, which gives rise to remarkable phenomena, such as the famous "glitches"
(sudden increases of the neutron star's rotation frequency) and changes in the star's cooling behaviour. However, making reliable predictions for the superfluid critical temperature
and for the "entrainment" between the neutron gas and the clusters remains a challenging problem for nuclear many-body theory. From a theoretical perspective, there are some analogies
between the neutron gas and ultracold Fermi gases in atom traps, which may help to make theoretical predictions for the superfluid neutron gas more reliable.