  We discuss how to generate random unitary matrices from the classical compact
groups U(N), O(N) and USp(N) with probability distributions given by the
respective invariant measures. The algorithm is straightforward to implement
using standard linear algebra packages. This approach extends to the Dyson
circular ensembles too. This article is based on a lecture given by the author
at the summer school on Number Theory and Random Matrix Theory held at the
University of Rochester in June 2006. The exposition is addressed to a general
mathematical audience.
