A moment map is part of the definition of a Hamiltonian group action on a symplectic manifold. I will talk about a version of the well-known Duistermaat-Heckman theorem and how the proof uses Chern-Weil theory. This theorem gives understanding of the symplectic form of the reduced space over an interval of regular values for the moment map.
Some constructions of morphisms in algebraic geometry generalise to symplectic manifolds, for example one can always blow up a symplectic submanifold (this was originally observed by Gromov). Roughly, you can do this by forming a seemingly ugly higher-dimensional symplectic manifold with a Hamiltonian group action; the reduced space turns out to be what you want.
Furthermore, Guillemin and Sternberg showed that when we pass through critical value for the moment map, reduced spaces are related by exactly such generalised “birational transformations”. I will try to explain their result in the most simple case of Hamiltonian circle actions and discuss some examples.