Integrable systems with elliptic dependence on momenta and related topics

We discuss a family of integrable many-body systems of classical (and quantum) mechanics. Some interrelations (dualities) predict existence of integrable many-body systems with elliptic dependence on particles momenta – the most general representative of this family. We describe some recent results on this topic. Next, we discuss relations of the many-body systems to other families of integrable models including integrable tops and spin chains. Finally, some interesting open problems are formulated.