Abstract: We will concentrate on the following two analytic problems of the random matrix theory. The first one is the evaluation of the large N (N-size of the matrix) limit of the basic eigenvalue statistics. The second one is the study of the analytic properties of the limiting distribution functions appeared after the large N limit . We will show that the both problems can be treated in the framework of the Riemann-Hilbert method of the theory of integrable systems. (No prior knowledge of either random matrices or integrable systems is needed)