pects of the Random Matrix Theory. First of all, the wealth of information obtained by heuristic methods of theoretical physics called for a proper understanding by mathematical standards. To that end, considerable efforts were directed towards proving conjectured universality  of correlations between eigenvalues of random matrices beyond classical (Gaussian or Circular) Wigner-Dyson ensembles, that is independence of their properties of the choice of the distribution of matrix elements for general RMT ensembles in the limit of large dimension. First, such a universality was verified for a rather general class of invariant ensembles , the line of research resulting, in particular, in development of the powerful Riemann-Hilbert approach to the asymptotics of orthogonal polynomials . More recently similar universality was extended also to a broad class of ensembles with independent, identically distributed entries . Deserve mentioning also works  on the large deviation statistics of spectral measure and on distribution of the number of eigenvalues in a given interval of spectrum, as well as the paper  which nontrivially extends the Dyson work on Brownian motion of eigenvalues by relating it to non-intersecting random walks.
Around the same time the statistics of the largest eigenvalue in Gaussian Ensembles attracted considerable attention . Its probability density first derived by C. Tracy and H. Widom  appeared to be highly universal  and emerged in such important combinatorial problems as the distribution of the length of the longest increasing subsequence of random permutations  as well as in applications in statistical mechanics: statistics of the height of random surfaces obtained by polynuclear growth  and the lowest energy state as well as the free energy of a directed polymer in random environment . Yet another highly influential development followed the works by Keating and Snaith  in providing powerful evidences in favour of a very intimate connection between the properties of characteristic polynomials of random matrices and the moments of the Riemann zeta-function (and other L-functions) along the so-called critical line. That line of research stimulated interest in general correlation properties of the characteristic polynomials of RMT ensembles . Ideas coming from the field of numerical matrix analysis lead to a fruitful and promising enrichment of the Random Matrix theory by "beta-ensembles" due to Dumitriu and Edelman  with a continuous positive parameter beta characterizing the statistics of their eigenvalues, the three discrete values beta=1,2 and 4 corresponding to the classical Orthogonal, Unitary, and Symplectic RMT ensembles. Apart from that, a steady growth of attention to various aspects of random matrix ensembles of non-Hermitian matrices with eigenvalues scattered in the complex plane  is to be mentioned, extending and generalizing early works by Ginibre  and Girko . Among other actively researched topics deserve mentioning works on singular values distributions and eigenvalues of random covariance matrices , important, in particular, for applications in quantum information context  and for the analysis of multivariate data in time series appearing in financial mathematics . Actively investigated were also random matrix ensembles with multifractal eigenvectors and/or the so-called critical eigenvalue statistics , ensembles of heavy-tailed random matrices , of sparse random matrices , and of Euclidean random matrices , as well as random matrices in external source and coupled in a chain . Increasingly important role in many RMT developments continue to play Harish-Chandra-Itzykson-Zuber integration formula  and its extensions, as well as the Selberg Integral, see  for a recent review.
- Wishart, J. Generalized product moment distribution in samples. Biometrika 20 A (1928) no. 1/2, 32-52 [Correction: ibid. 20 A.(1928) p.425]
- James, A. T. Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. v. 25 (1954) no.1, 40-75; James, A. T. The Distribution of the Latent Roots of the Cova