  We review the quantum statistical properties of two-dimensional shell-shaped
gases, produced by cooling and confining atomic ensembles in thin hollow
shells. We consider both spherical and ellipsoidal shapes, discussing at zero
and at finite temperature the phenomena of Bose-Einstein condensation and of
superfluidity, the physics of vortices, and the crossover from the
Bardeen-Cooper-Schrieffer regime to a Bose-Einstein condensate. The novel
aspects associated to the curved geometry are elucidated in comparison with
flat two-dimensional superfluids. We also describe the hydrodynamic excitations
and their relation with the Berezinskii-Kosterlitz-Thouless transition for
two-dimensional flat and curved superfluids. In the next years, shell-shaped
atomic gases will be the leading experimental platform for investigations of
quantum many-body physics in curved spatial domains.
