In symplectic geometry, the moment map (or momentum map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic quotients, symplectic cuts and sums. During the lecture I will introduce this concept and I will show you how to construct symplectic manifolds having some specific image for the moment map. Also I'll say something about the relation between combinatorial data in the image of the moment map and geometric data on the manifold