In this talk we will investigate some topological properties of the space Fk(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces Fk(M) to become increasingly complicated. Church and others showed, however, that when M is a connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and how they generalize classical notions of homological stability proved by McDuff and Segal in the 1970s. I will describe higher-order ``secondary stability'' phenomena established in recent work joint with Jeremy Miller. The project is inspired by work of Galatius--Kupers--Randal-Williams.