Chip-firing processes are discrete dynamical systems. A commodity (chips, sand, dollars) is exchanged between sites of a network according to simple local rules. Although governed by local rules, the long-term global behavior of the system reveals unexpected properties, including intricate fractal-like patterns. Early results related chip-firing to classic combinatorial objects such as spanning trees, parking functions, and matroids. In recent years, chip-firing has seen much activity in new directions. Connections have been made, for example, between chip-firing and Coxeter groups, binomial ideals, and Riemann surfaces. In this talk, I will give a broad survey of the theory of chip-firing and its many ties to algebraic combinatorics.