  Since the first experimental realization of Bose-Einstein condensation in
cold atomic gases in 1995 there has been a surge of activity in this field.
Ingenious experiments have allowed us to probe matter close to zero temperature
and reveal some of the fascinating effects quantum mechanics has bestowed on
nature. It is a challenge for mathematical physicists to understand these
various phenomena from first principles, that is, starting from the underlying
many-body Schr\"odinger equation. Recent progress in this direction concerns
mainly equilibrium properties of dilute, cold quantum gases. We shall explain
some of the results in this article, and describe the mathematics involved in
understanding these phenomena. Topics include the ground state energy and the
free energy at positive temperature, the effect of interparticle interaction on
the critical temperature for Bose-Einstein condensation, as well as the
occurrence of superfluidity and quantized vortices in rapidly rotating gases.
