  Discrete subgroups of SL(2,R) are well understood, and classified by the
geometry of the corresponding hyperbolic surfaces. Discrete subgroups of
higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more
mysterious. While lattices in this setting are rigid, there also exist more
flexible, "thinner" discrete subgroups, which may have large and interesting
deformation spaces, giving rise in particular to so-called higher Teichm\"uller
theory. We survey recent progress in constructing and understanding such
discrete subgroups from a geometric and dynamical viewpoint.
