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.

58 pages, 16 figures, accepted version