  This paper is based on three tutorial lectures given at the CRM Summer School
in Banff, Alberta, June 27-July 10, 1999 on "Theoretical Physics at the End of
the Twentieth Century". They will be published by Springer-Verlag. Lecture I
gives a quick overview of recent studies of BEC in trapped atomic gases and
reviews some properties of a trapped non-interacting Bose gas. Lecture II
mainly discusses the collective modes of a pure condensate, based on the T=0 GP
equation of motion. The Stringari quantum hydrodynamic theory is introduced.
Lecture III discusses the coupled dynamics of the condensate and non-condensate
at finite temperatures. A recent derivation of the two-fluid hydrodynamic
equations is sketched. This theory predicts a new relaxational mode associated
with the fact that the condensate and non-condensate may not be in diffusive
equilibrium.
