Multiloop amplitudes in superstring theory.

String theory is a candidate for a unified theory of all forces in nature. String theory is a quantum theory, and, because it includes a state that can be identified with the graviton, it is a quantum theory of gravity. So it could be a framework for describing the four fundamental interactions. The main idea is to replace the description of particles as zero dimensional object with one dimensional object, a string. Mimicking the procedure followed in QFT one can formulate the theory using the Feynmann path integral. In this context one would be able to compute the amplitudes for a process of string scattering, as, for example, in QED. It was shown that correct definition of the amplitudes is strictly connected with the construction of a suitable measure on the moduli space of certain Riemann surfaces of genus g. Thus an essential element in the computation of string amplitudes is the definition of this chiral measure. During the last fifteen years the correct expression for the superstring measure was found just up to genus two. In this thesis I will present a general method to compute the measure up to five loop which as a byproduct also improves the known results for genus two and shows that the previous formulated ansatze for the genus three amplitude was wrong. In the first part I present the well known construction of the partition function in the bosonic theory. Actually, this is the starting point for the the generalization to the supersymmetric case. Then, I introduce the mathematical background we need to obtain the superstring measure. Finally, I discuss in great details the construction of the measure up to genus five and some open problems are emphasized. Moreover, I will also point out that the path integral procedure leads to some ambiguities related to the infinite dimension of the functional measure appearing in the path integral and I will explain the main difficulties related to its mathematically rigorous definition.