Over the course of the last century, a number of mathematicians have proved results suggesting a profound and subtle connection between the apparently disparate fields of topology and number theory. In essence, these results say that the solutions to polynomial equations are controlled by the topology of the graph of those equations. In this talk, I will describe a fundamental idea of Alexander Grothendieck, which makes precise what this mysterious connection should be, and outline some of my own recent work which sheds further light on this theory.