  We survey recent work on the dynamics of the outer automorphism group of a
word hyperbolic group on spaces of (conjugacy classes of) representations ofthe
group into a semi-simple Lie group G. All these results are motivated by the
fact that the mapping class group of a closed surface acts properly
discontinuously on the Teichmuller space of the surface.
  We explore two settings. 1) The work of Guichard, Labourie and Wienhard
establishing the proper discontinuity of the action of outer automorphism
groups on spaces of Anosov representations. 2) The work of Canary, Gelander,
Lee, Magid, Minsky and Storm on the case where the domain group is the
fundamental group of a compact 3-manifold with boundary and G=PSL(2,C).
