Convex cocompact subgroups of rank-one semisimple Lie groups (such as PSL(2,R)) are prototypical examples of hyperbolic groups, and form a structurally stable class of quasi-isometrically embedded discrete subgroups. Anosov subgroups of higher rank groups (such as PSL(n,R) with n>2) are an analogue of convex cocompact subgroups of rank-one groups which share many of their good geometric and dynamical properties. I will introduce a higher-rank analogue of geometric finiteness, which may be thought of as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity, and describe some of its geometric and dynamical consequences. Speaker(s): Feng Zhu (University of Michigan)