  Recent years have seen very important developments at the interface of
Diophantine approximation and homogeneous dynamics. In the first part of the
paper we give a brief exposition of a dictionary developed by Dani and
Kleinbock-Margulis which relates Diophantine properties of vectors to
distribution of orbits of flows on the space of unimodular lattices. In the
second part of the paper we briefly describe an extension of this dictionary
recently developed by the authors, which establishes an analogous dynamical
correspondence for general lattice orbits on homogeneous spaces. We concentrate
specifically on the problem of estimating exponents of Diophantine
approximation by arithmetic lattices acting on algebraic varieties. In the
third part of the paper, we exemplify our results by establishing explicit
bounds for the Diophantine exponent of dense lattice orbits in a number of
basic cases.
  These include the linear and affine actions on affine spaces, and the action
on the variety of matrices of fixed determinant. In some cases, these exponents
are shown to be best possible.
