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.