This short note serves as a joint introduction to the papers "On Actions of Epimorphic Subgroups on Homogeneous Spaces" by Nimesh Shah and Barak Weiss (Stony Brook IMS preprint 1999/7b) and "Unique Ergodicity on Compact Homogeneous Spaces" by Barak Weiss. For the benefit of the readers who are not experts in the theory of subgroup actions on homogeneous spaces I have prefaced the papers with some general remarks explaining and motivating our results, and the connection between them. The remarks are organized as a comparison between facts which had been previously known about the action of the geodesic and horocycle flow on finite-volume Riemann surfaces -- the simplest nontrivial example that falls into our framework -- and our results on subgroup actions on homogeneous spaces.