A Kleinian group is a discrete subgroup of the orientation-preserving isometry group of hyperbolic 3-space. The boundary of hyperbolic 3-space the two-sphere. A Kleinian group has a limit set on the two-sphere which is the set of limit points of all orbits of points in hyperbolic space. I will discuss a structure theorem which allows one to determine some of the structure of the associated hyperbolic manifold from the combinatorics of the limit set of a geometrically finite Kleinian group. Namely, one can recognize the decomposition of the characteristic sub-manifold.

I will also discuss a generalization of this structure theorem to hyperbolic groups. These are groups which act geometrically on a $\delta$-hyperbolic metric space. They have a naturally defined boundary, which is analogous to the limit set. I'll also discuss several open questions.