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.