Let $H$ be a subgroup of $G$ generated by unipotents, and consider
its action on $\Gamma \backslash G$.  Building on the work of Ratner a
theorem of Mozes and Shah shows that limits of $H$-invariant and
ergodic probability measures are again ergodic and therefore Haar
measures on closed orbits. In joint work with Margulis and Venkatesh
we prove (under certain technical conditions) a rate of convergence in
this theorem if the acting group is semisimple. We will discuss the
theorem and some of its ideas which includes a sketch of a proof of a
special case of Ratner's theorem on invariant measures.