Algebraic topology was invented by Poincare in 1895 to study the behavior of algebraic functions.  In his seminal ICM address 5 years later, Hilbert posed a fundamental challenge to the field: find a topological obstruction to reducing the solution of the general degree 7 polynomial to an expression in functions of two or fewer variables.  In this talk, I'll review some of the beautiful history of algebraic topology and algebraic functions, discuss Hilbert's problem(s), and outline ongoing work in applying the topology of braids and algebraic functions to this problem.  This is joint work with Benson Farb.