Some of the remarkable phenomena that emerge when a trapped, ultracold Bose gas is set in rapid rotational motion will be reviewed. In anharmonic traps, where the rotational velocity can in principle be arbitrarily large, one can distinguish three critical velocities at which the flow pattern changes radically. The first is the velocity at which vorticity sets in, eventually leading to a lattice of vortices, at the second a 'hole' is created and the condensate becomes concentrated in an annulus while the vortex lattice persists in the bulk, and at the third a transition to a 'giant vortex' state takes place in which all vorticity disappears from the bulk but a macroscopic circulation around the hole remains.The mathematical model used for analysis of these phenomena has similarities with Ginzburg-Landau (GL) Theory in superconductivity with the critical velocities in rotating gases playing an analogous role to the critical magnetic fields in GL theory, and techniques originally developed in the context of GL theory have, indeed, been important for understanding rotating gases. There are, however, also important differences, and in particular the theory of the giant vortex transition requires significant modifications of the GL setting. Theses similarities and differences will be discussed. (Joint work with Michele Correggi and Nicoals Rougerie.)