I will discuss questions and results related to polynomials in a large number of variables, starting with classical results of Hilbert and moving to Stillman’s Conjecture and its proof by Ananyan—Hochster.  Then I will describe ways to think about the limit of polynomial rings as the number of variables goes to infinity, and how this can be applied to obtain new finiteness results.  The original work covered in the talk is all joint with Steven V Sam and Andrew Snowden.