  Atomic gases confined in curved geometries are characterized by distinctive
features that are absent in their flat counterparts, such as periodic
boundaries, local curvature, and nontrivial topologies. The recent experiments
with shell-shaped quantum gases and the study of ring-shaped superfluids point
out that the manifold of a quantum gas could soon become a controllable
feature, thus allowing to address the fundamental study of curved many-body
quantum systems. Here, we review the main geometries realized in the
experiments, analyzing the theoretical and experimental status on their phase
transitions and on the superfluid dynamics. In perspective, we delineate the
study of vortices, the few-body physics, and the search for analog models in
various curved geometries as the most promising research areas.
