Hurwitz proved that a complex curve of genus $g>1$ has at most $84(g-1)$ automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups $G$ behave the same way: the least $g>1$ for which $G$ acts on a genus-$g$ curve is on the order of $(G)\times d(G)$, where $d(G)$ is the minimal number of generators of $G$. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from others. If we restrict to $G$-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.