The aim of this seminar is to give a short introduction to the basic theory of tranformation groups giving relevant applications in Riemannian geometry. In particular we present Hamiltonian actions discussing some recent results like uniqueness and multiplicty of symplectic and Lagrangian orbits, the convex envelope of a coadjoint orbit and how can we estimate the first eigenvalue of a compact Hermitian symmetric space. Some results were obtained in collaboration with Dr. Alessandro Ghigi.