  This article reviews developments in the theory of rapidly rotating
degenerate atomic gases. The main focus is on the equilibrium properties of a
single component atomic Bose gas, which (at least at rest) forms a
Bose-Einstein condensate. Rotation leads to the formation of quantized vortices
which order into a vortex array, in close analogy with the behaviour of
superfluid helium. Under conditions of rapid rotation, when the vortex density
becomes large, atomic Bose gases offer the possibility to explore the physics
of quantized vortices in novel parameter regimes. First, there is an
interesting regime in which the vortices become sufficiently dense that their
cores -- as set by the healing length -- start to overlap. In this regime, the
theoretical description simplifies, allowing a reduction to single particle
states in the lowest Landau level. Second, one can envisage entering a regime
of very high vortex density, when the number of vortices becomes comparable to
the number of particles in the gas. In this regime, theory predicts the
appearance of a series of strongly correlated phases, which can be viewed as
{\it bosonic} versions of fractional quantum Hall states. This article
describes the equilibrium properties of rapidly rotating atomic Bose gases in
both the mean-field and the strongly correlated regimes, and related
theoretical developments for Bose gases in lattices, for multi-component Bose
gases, and for atomic Fermi gases. The current experimental situation and
outlook for the future are discussed in the light of these theoretical
developments.
