This talk will illustrate some topological properties of the space F_k(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 F_k(M) to become increasingly complicated. Church and others showed, however, that when M is 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 describe a higher-order “secondary representation stability” phenomenon among the unstable homology classes. These results may be viewed as a representation-theoretic analogue of current work of Galatius–Kupers–Randal-Williams. The project is joint with Jeremy Miller.