Recent work by Baker, Norine, and others has drawn parallels between the theory of divisors on graphs and results in algebraic geometry. In this talk, we will review some of the ways that these theories are -- and aren´t -- analogous, and in particular discuss some recent joint work with Criel Merino about a theorem of Kani and Rosen on the decomposition of Jacobians of curves that have certain automorphism groups and the way this theorem translates to the graph theoretic setting.