  This expository paper introduces the theory of Harish-Chandra integrals, a
family of special functions that express the integral of an exponential
function over the adjoint orbits of a compact Lie group. Originally studied in
the context of harmonic analysis on Lie algebras, Harish-Chandra integrals now
have diverse applications in many areas of mathematics and physics. We review a
number of these applications, present several different proofs of
Harish-Chandra's celebrated exact formula for the integrals, and give detailed
derivations of the specific integral formulae for all compact classical groups.
These notes are intended for mathematicians and physicists who are familiar
with the basics of Lie groups and Lie algebras but who may not be specialists
in representation theory or harmonic analysis.
