Bose-Einstein condensates in fast rotation

In this short review we present our recent results concerning the rotation of atomic Bose-Einstein condensates confined in quadratic or quartic potentials, and give an overview of the field. We first describe the procedure used to set an atomic gas in rotation and briefly discuss the physics of condensates containing a single vortex line. We then address the regime of fast rotation in harmonic traps, where the rotation frequency is close to the trapping frequency. In this limit the Landau Level formalism is well suited to describe the system. The problem of the condensation temperature of a fast rotating gas is discussed, as well as the equilibrium shape of the cloud and the structure of the vortex lattice. Finally we review results obtained with a quadratic + quartic potential, which allows to study a regime where the rotation frequency is equal to or larger than the harmonic trapping frequency.