A divisor on a graph is an assignment of integer weights to each of the vertices. One assignment is considered linearly equivalent to another assignment if it can be transformed into it by a series of legal moves called chip-firing moves. The Picard group of a graph, named based on its relationship to the Picard group of a scheme, is defined as the free abelian group of divisors on the graph up to linear equivalence. This group will be the central focus of the talk, and we'll compute Pic(G) for some classes of graphs G. We'll also define the rank of a divisor and a related graph invariant called gonality.