Topology began with the study of complex functions, and the interaction of topology with algebraic geometry and number theory continues to be fertile ground.  Starting with basic examples, I will explain how the function field/number field dictionary and the remarkable framework of the Weil conjectures (as established by Weil, Grothendieck, Dworkin, Deligne and many others) allows one to use counting problems to predict and discover topological properties of manifolds, and vice versa. This is joint work with Benson Farb and Melanie Matchett Wood.