  In this paper, we provide a complete theory of Diophantine approximation in
the limit set of a group acting on a Gromov hyperbolic metric space. This
summarizes and completes a long line of results by many authors, from
Patterson's classic '76 paper to more recent results of Hersonsky and Paulin
('02, '04, '07). Concrete examples of situations we consider which have not
been considered before include geometrically infinite Kleinian groups,
geometrically finite Kleinian groups where the approximating point is not a
fixed point of the group, and groups acting on infinite-dimensional hyperbolic
space. Moreover, in addition to providing much greater generality than any
prior work of which we are aware, our results also give new insight into the
nature of the connection between Diophantine approximation and the geometry of
the limit set within which it takes place. Two results are also contained here
which are purely geometric: a generalization of a theorem of Bishop and Jones
('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial
limit set of a group acting on a proper geodesic Gromov hyperbolic metric space
has zero Patterson--Sullivan measure unless the group is quasiconvex-cocompact.
The latter is an application of a Diophantine theorem.
