Random walks are well understood by now. However, if we
require a random walk not to intersect itself, so that it is a self-avoiding
walk, then it is much more difficult to analyse and many of the important
mathematical problems remain unsolved. This lecture will give an overview of
some of what is known about self-avoiding walks, including some old and some
more recent results, using methods that touch on combinatorics, probability, and
statistical mechanics.