In characteristic zero, a curve of genus g>1 has at most 84(g-1) automorphisms. In characteristic p>0 this is no longer true, e.g. the degree-(1+p^n) Fermat curves violate this bound. I will discuss the known results on curves with many automorphisms (in positive characteristic), and in particular I will present the first positive-dimensional families of such curves. Precisely, for each of infinitely many positive integers g, I will present an infinite collection of non-isomorphic genus-g curves each of which has more than 84(g-1) automorphisms. (Joint work with Bob Guralnick, Peter Mueller, and Joel Rosenberg.)