The Abelian sandpile is a diffusion process on configurations of chips on the integer lattice, in which a vertex with at least 4 chips can "topple", distributing one of its chips to each of its 4 neighbors. Though the sandpile has been the object of study from a diverse set of perspectives, even some of the most basic questions about the terminal configurations produced by the process have remained unanswered. One of the most striking features of the sandpile is that when begun from a large concentration of n chips, the resulting terminal configurations seem to converge to a peculiar fractal pattern as n goes to infinity. In this talk, we will discuss a mathematical explanation for the fractal nature of the sandpile which comes from a peculiar conjecture connecting Apollonian circle packings to integer superharmonic functions on the lattice. Time permitting, we will discuss the path to a proof of the conjecture. (Joint work with Charles Smart and Lionel Levine.)