The study of evolutionary dynamics was so far mainly restricted to finite strategy spaces. In this paper we show that this restriction is in most cases unnecessary. We give a mild condition under which the continuous time replicator dynamics are well defined for infinite strategy spaces. Furthermore, we provide conditions for the stability of rest points. Finally, we apply this general theory to a number of applications like the Nash demand game, the War of Attrition, Cournot and Bertrand oligopoly games, and mixed strategies.