Let G be a Lie group, X be a G-homogeneous space
of finite volume, and H be a closed subgroup of G.
What are the H-invariant probabilities on X?
What are the H-invariant closed subsets in X?
I will first survey Ratner's results addressing
the case when H is generated by unipotent elements.
I will then focus on a joint work with J.F. Quint
addressing the case when G is simple and
H is Zariski dense. In this case,
the Haar probability on X is the only
atom-free H-invariant probability on X.
Moreover every H-invariant
subset of X is either finite or dense.
The proof uses random walks on X.