  Given a lattice \Gamma in a locally compact group G and a closed subgroup H
of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an
increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma,
v\in V, and compactly supported function \phi on V, we consider the sums
  S_{\phi,v}(T)=\sum_{\gamma\in \Gamma_T} \phi(v \gamma).
  Understanding the asymptotic behavior of S_{\phi,v}(T) is a delicate problem
which has only been considered for certain very special choices of H, G and
{\Gamma_T}. We develop a general abstract approach to the problem, and apply it
to the case when G is a Lie group and either H or G is semisimple. When G is a
group of matrices equipped with a norm, we have
  S_{\phi,v}(T) \sim \int_{G_T} \phi(vg) dg, where G_T={g\in G:||g||<T} and
\Gamma_T = G_T \cap \Gamma. We also show that the asymptotics of S_{\phi,v}(T)
is governed by \int_V \phi d\nu, where \nu is an explicit limiting density
depending on the choice of v and the norm.
