Topology, Geometry and Analysis on coadjoint orbits.

A new mathematical object appeared in connection
to the Orbit method is the notion of
a coadjoint orbit of a Lie group G in the space g* dual
to the Lie algebra Lie(G).
It turned out that topology and geometry
of these orbits are very important in representation
theory of Lie groups and modern mathematical physics
(integrable systems, moment map,
symplectic reduction etc).
I explain main facts and open problems for non-experts.