In this thesis we study the classical and quantum momentum maps
and the theory of reduction. We focus on the notion of momentum
map in Poisson geometry and we discuss the classification of the
momentum map in this framework. Furthermore, we describe the
so-called Poisson Reduction, a technique that allows us to reduce
the dimension of a manifold in presence of symmetries implemented
by Poisson actions.
Using techniques of deformation quantization and quantum
groups, we introduce the quantum momentum map as a deformation
of the classical momentum map, constructed in such a way
that it factorizes the quantum action. As an application we discuss
some examples of quantum reduction.