One of the key issues related to superfluidity is the existence of vortices. In very recent experiments on Bose–Einstein condensates, vortices have been observed by rotating the trap holding the atoms. In contrast to a classical fluid for which the equilibrium velocity corresponds to solid body rotation, a quantum fluid such as a Bose–Einstein condensate can rotate only through the nucleation of quantized vortices. This monograph is dedicated to the mathematical modelling of these phenomena.The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. The mathematical analysis is made in the framework of the Gross–Pitaevskii energy. Results are presented and open problems related to recent experiments are explained.The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.