This is a collection of expository articles about various topics at the interface between enumerative combinatorics and stochastic processes. These articles expand on a course of lectures given at the Ecole d’Eté de Probabilités de Saint-Flour XXXII in July 2002. Each article is fairly self-contained, so readers with adequate background can start reading any article, with occasional consultation of earlier articles as necessary. Most articles include both exercises and open problems