Moduli spaces are important objects in mathematics. They are used to endow the set of isomorphy classes of objects, structures, etc. with a geometric structure. They are not only of relevance in mathematics but also in physics, where they play a prominent role in the context of quantization. One of the basic principles of quantization is the weighted summation over all possible configurations of a system, thereby obtaining configurations spaces.If one divides out symmetries one ends up with the moduli space.To make mathematical sense out of this procedure is already quite challenging. There not always exists exact mathematical results. The goal of this mathematical proposal is to advance the mathematical knowledge with regard to studying properties of moduli spaces of relevance in quantization and to go one step further to quantize these moduli spaces. In this respect one of the main motivations of the proposal is to participate in the goal to make quantum field theory a mathematicalentity.In this project we deal also with the quantization of singular spaces and with the singularities of moduli spaces. Models of Topological Quantum Field Theory (Chern-Simons Theory) and Conformal Field Theory (Wess-Zumino-Novikov-Witten models), Feynman integrals and new approaches to an algebraization of 3-manifolds will show up.From the very nature of the moduli space the approaches used are geometric in nature. Modern geometry (algebraic geometry, Kähler geometry, symplectic geometry) shows up but also operator theory and the algebras of Conformal Field Theory will appear. The relation between mathematics and physics is not a one-way street only in direction of physics. On the contrary, very important results in mathematics weretriggered by methods which had their heuristic origin in physics.