Large deviations theory aims to estimate the probability of rare events. For instance it allows to estimate the probability that after throwing a fair dice 100 times, we observed the number 1 50 times, whereas law of large numbers would have predict rather less than 20 times. The classical results from large deviations do not apply to complicate functions of independent variables such as the eigenvalues of random matrices. These mathematical objects appear in the analysis of large arrays of data as first noticed by Wishart, in theoretical physics since the work of Wigner or questions in statistical learning. During the last twenty years, important advances allowed to analyse the large deviations for very particular models of random matrices, but a full theory is still lacking. The goal of this project is to develop such a theory.