This talk will introduce you to several instances of a recurring theme
in topology and dynamics: that the groups Diff(M) and Homeo(M) of
self-diffeomorphisms or homeomorphisms of a compact, smooth manifold
behave in many ways like finite dimensional semi-simple Lie groups.
This occurs both on the global, structural level, and also when
considering spaces of representations of finitely generated groups
into these larger groups. I will discuss some classical motivation
and some of my recent results on this theme, and explain some of the
broader (geometric, topological, and dynamical) motivation to study
these linear and non-linear representations.