  After reviewing the ideal Bose-Einstein gas in a box and in a harmonic trap,
I discuss the effect of interactions on the formation of a Bose-Einstein
condensate (BEC), along with the dynamics of small-amplitude perturbations (the
Bogoliubov equations). When the condensate rotates with angular velocity Omega,
one or several vortices nucleate, with many observable consequences. With more
rapid rotation, the vortices form a dense triangular array, and the collective
behavior of these vortices has additional experimental implications. For Omega
near the radial trap frequency omega_perp, the lowest-Landau-level
approximation becomes applicable, providing a simple picture of such rapidly
rotating condensates. Eventually, as Omega approaches omega_perp, the rotating
dilute gas is expected to undergo a quantum phase transition from a superfluid
to various highly correlated (nonsuperfluid) states analogous to those familiar
from the fractional quantum Hall effect for electrons in a strong perpendicular
magnetic field.
