The pentagram map, introduced by Schwartz, is a discrete-time dynamical system on polygons in the projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. We show that, over any algebraically closed field of characteristic not equal to 2, the pentagram defines a discrete integrable system, meaning that it is birational to a translation on a family of abelian varieties. This makes it possible to understand the finite field dynamics of the pentagram map. In this talk, we will explain this result and discuss its intersection with some central open problems in the dynamics of rational maps over finite fields.