Hilbert's 13th Problem is a fundamental open problem about 1-variable polynomials.  In this talk I will explain how Hilbert's 13th (and related problems) fits into a wider framework that includes, for example, 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.  Most of this talk should be accessible to undergraduates.