The divergence function provides a way to measure some of the large-scale curvature of a group or metric space by quantifying how quickly pairs of geodesic rays separate from each other.  Questions about the study of divergence functions were initially raised by Gromov and have received significant attention in the past several years.  This talk will provide an introduction to this invariant and its use in geometric group theory.  We will also discuss our proof of a threshold theorem for divergence in random groups, which we proved by establishing an Erdos-Renyi style threshold theorem in random graphs.