A p-local finite group consists of a finite p-group S, together with a pair of categories that encode "conjugacy relations" among the subgroups of S, modelled on the fusion in Sylow p-subgroups of finite groups. To any p-local finite group, we associate a classifying space. These spaces share many of the same properties of p-completed classifying spaces of finite groups. In particular, a p-local finite group is characterized by its classifying space. This is a survey talk on the theory, from motivation to recent results.