Abstract: In the 19th century, Emmy Noether was the first to relate invariance, in a dynamical system, with constants of motions. The modern version of this theorem, with the coming of symplectic formalism in classical (or quantum) mechanics, involves the so-called Moment Map. This moment map is associated to every symplectic or pre-symplectic manifold together with an invariant action of a Lie group. It is defined on the manifold with values in the dual of the Lie algebra of the group of symmetries. Souriau's version of Noether's theorem states that: the moment map is constant on the characteristics of the pre-symplectic form. But the role of this moment map has increased in parallel with the development of symplectic geometry, it became the fundamental tool for classification theorems in symplectic geometry, in presence of symmetries. The classical version of the moment map is now well established, since the beginning of the 70's. But in the recent decades, some objects looking like moment maps appeared in other contexts, not covered by the classical formalism: spaces regarded as singular in the classical framework of differential geometry, orbifolds, quotients by symplectic reductions, infinite dimensional modular spaces etc. It was interesting to try to give to these many recent heuristic examples and constructions a unique simple and efficient framework.