Hilbert's 13th Problem (H13) is a fundamental open problem about polynomials in one variable.  It is part of a beautiful (but mostly forgotten) story going back 3 thousand years.  In this talk I will explain how H13 (and related problems) fits into a wider framework that includes problems in enumerative algebraic geometry and the theory of modular functions. I will then report on some recent progress, joint with Mark Kisin and Jesse Wolfson.  While some fancy objects will appear in this talk, much of it should (I hope) be understandable to undergraduate math majors.