The principal goal of Random Matrix Theory (RMT) is the description of the statistical properties of the eigenvalues or singular values of ensembles of matrices with random entries subject to some chosen distribution, in particular when the size of the matrix becomes very large. The statistical distributions that occur in this regime of large sizes display some features which are very robust in the sense that they appear rather independently of the distribution chosen for the matrix entries. This phenomenon goes under the general headname of universality and it is not conceptually dissimilar from the more commonly known central limit theorem. The mathematics required by or developed for RMT has found or has come from a stunningly wide range of areas in both mathematics an theoretical physics, such as for example, approximation theory, orthogonal polynomials and their asymptotics, combinatorics, dynamical systems of integrable type, representation theory of finite and infinite groups, quantum gravity, conformal field theory, string theory just to name a few.
The notion of universality appears prominently also in the small-dispersion limit of integrable nonlinear waves, for example the Korteweg-deVries, nonlinear Schrödinger equations, etc.
The journal SIGMA is preparing to make a special issue in honor of Percy Deift and Craig Tracy. See the following link regarding how to submit a paper.