These lecture notes study the interplay between randomness and geometry of graphs. The first
part of the notes reviews several basic geometric concepts before moving on to examine the
manifestation of the underlying geometry in the behavior of random processes mostly
percolation and random walk. The study of the geometry of infinite vertex transitive graphs
and of Cayley graphs in particular is fairly well developed. One goal of these notes is to
point to some random metric spaces modeled by graphs that turn out to be somewhat exotic that
is they admit a combination of properties not encountered in the vertex transitive world.
These include percolation clusters on vertex transitive graphs critical clusters local and
scaling limits of graphs long range percolation CCCP graphs obtained by contracting
percolation clusters on graphs and stationary random graphs including the uniform infinite
planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).
