Let G be a complex semisimple Lie group, and let M be a topological space. The G-character variety of S, roughly speaking, is the space of conjugacy classes of representations from pi_1(M) to G. In the case where M is a surface, we get an action of Out(pi_1(M)) on the character variety. We will survey various results about this action, with the following motivating question: to what extent does Out(pi_1(M)) act properly discontinuously on the character variety? This talk will involve some topology, some algebra, and some analysis, which will likely either satisfy or upset the entire audience.