In the first part of the talk, I'll review the classical theory of coadjoint orbits including the Kirillov-Kostant-Souriau symplectic structure, Duistermaat-Heckman oscillating integrals and equivariant localization.

In the second part, I'll talk about coadjoint orbits of Virasoroalgebra, the Schwarzian action and infinite dimensional orbital integrals. If time permits, we will touch upon recent applications of coadjoint orbits to ideas of holography in physics.

The talk is based on a joint work with S. Shatashvili.