A group G is said to commensurate a subgroup H, if for all g \in G, H^g \cap H is of finite index in H and H^g, where H^g denotes the conjugate of H by g. The commensuration action of G on H can be studied dynamically. This gives rise to two extreme behaviors: hyperbolic and elliptic. We will discuss what these mean and survey a range of theorems and conjectures in this context, starting with work of Margulis, and coming to the present day.
