Focusing on a number of problems related to the intersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics covered include: discrete harmonic measure; the probability that independent random walks do not intersect; and properties of walks without self-intersections. With the inclusion of a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, this text should be of use to researchers in probability and statistical physics, and to graduate students interested in basic properties of random walk.