We consider questions about the automorphism groups of curves particularly over algebraically closed fields of positive characteristic. It is a classical result that such groups are finite as long as the genus is at least 2. Stichtenoth proved that Aut(X) is at most a degree 4 polynomial in the genus but these curves are quite rare. We will discuss a more group theoretic point of view towards this problem and some recent results particularly for ordinary curves. We will also discuss a result of Ritzenhaler about automorphism groups of the reduction of the modular curve and improvements of his result.