Asymptotic representation theory over $\mathbb Z$

Representation theory over $\mathbb Z$ is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover, it turns out that important questions in topology/number theory/representation theory/... correspond to asking whether familiar algebraic properties hold for these "rings". I'll explain how these connections work; describe what we know and don't know; and give a wide sampling of applications in different fields where this has led to concrete results. No knowledge of representation theory will be required---indeed, that's sort of the whole point!