  For a geometrically finite group Gamma of G=SO(n,1), we survey recent
developments on counting and equidistribution problems for orbits of Gamma in a
homogeneous space H\G where H is trivial, symmetric or horospherical. Main
applications are found in an affine sieve on orbits of thin groups as well as
in sphere counting problems for sphere packings invariant under a geometrically
finite group. In our sphere counting problems, spheres can be ordered with
respect to a general conformal metric.
