A polynomial functor P is a functor from the category of finite-dimensional vector spaces to itself such that for every U,V the map Hom(U,V) -> Hom(P(U),P(V)) is polynomial. In characteristic zero, P is a direct sum of Schur functors. This talk concerns closed subsets of such P, i.e., rules that assign to a vector space V a closed subset X(V) of P(V) such that P(phi)X(U) is contained in X(V) for every linear map phi:U -> V. Quite surprisingly, these behave very much like finite-dimensional affine varieties. For instance, they satisfy the descending chain condition and a version of Chevalley's theorem on constructible sets. I will discuss these results and more. The talk is based on joint work with Arthur Bik, Rob Eggermont, and Andrew Snowden.