  This is a rather personal review of the problem of self-avoiding walks and
polygons. After defining the problem, and outlining what is known rigorously
and what is merely conjectured, I highlight the major outstanding problems. I
then give several applications in which the I have been involved. These include
a study of surface adsorption of polymers, counting possible paths in a
telecommunication network, hitting probabilities of SAWs in a rectangle, and
the modelling of biological experiments on polymers. I hope to show that SAWs
are not only of intrinsic mathematical interest, but also have many interesting
and useful applications.
