A functor P from {finite-dimensional K-vector spaces} to itself is called polynomial if for each V,W, the map P:Hom(V,W)->Hom(P(V),P(W)) is polynomial. A subvariety X of P is the data of a closed subvariety X(V) of P(V) for every V, such that P(f) maps X(V) into X(W) for every f in Hom(V,W). Examples include Segre-Veronese varieties, their secant
varieties, and varieties of tensors of bounded slice rank. I will discuss the structure theory of varieties X in polynomial functors P: they are (topologically) Noetherian, admit a version of Chevalley's theorem on constructible sets, are unirational up to a finite-dimensional variety, admit Buchberger-type algorithms for elimination, and more. 

(Based on joint work with Arthur Bik, Rob Eggermont, Andrew Snowden, Alessandro Danelon, Emanuele Ventura, Andreas Blatter.)