es. Along those lines Wigner succeeded in calculating the simplest nontrivial spectral characteristics of random real symmetric matrices with independent, identically distributed entries - the mean density of eigenvalues, and demonstrated that in the limit of large matrix size it is given quite generally by the so called Semicircular Law. Wigner also provided insights into statistics of separations between the neighbouring eigenvalues of such matrices (Wigner surmise).
Boosted by those works RMT problems attracted considerable attention and within the next few years many essential tools helping to analyse properties of random matrices were developed, most notably the method of orthogonal polynomials by Mehta and Gaudin . Wigner's ideas were further substantiated by the seminal Dyson works  who gave important symmetry classification of Hamiltonians implying the existence of three major symmetry classes of random matrices - Orthogonal, Unitary and Symplectic, which cover the most relevant classical ensembles. Dyson also introduced the abovementioned circular versions of random matrix ensembles, developed a detailed theory of their spectra, and suggested a model of Brownian motion in random matrices ensembles which proved to be conceptually important and established a link to exactly soluble systems, such as the Calogero-Sutherland-Moser model.
Starting from the 1980's and further into 1990's the interest in RMT ideas, methods and results became widespread in the Theoretical Physics community. Motivated initially mainly by applications in Nuclear Physics (see  for an early influential review) RMT was further influenced by advances in the field of Quantum Chaos, such as the conjecture of Bohigas, Giannoni, and Schmit (BGS)  claiming statistical similarity between RMT spectra and highly excited energy levels of generic quantum systems whose classical counterparts show chaotic dynamical behaviour (see  for recent works towards justification of the BGS conjecture). From a different angle, somewhat similar ideas were substantiated and quantified by applications of powerful methods coming from the field of disordered systems and the Anderson localization, most importantly by supermatrix approach due to Efetov  and its further adaptation to RMT by Weidenmueller, Verbaarschot, and Zirnbauer  ( see  for a few recent reviews). Also influential were a highly successful maximal-entropy based RMT approach to statistics of electronic transport in quantum-coherent (mesoscopic) samples introduced by Mello and collaborators , as well as the idea of parametric dependence of spectra of disordered and chaotic systems, and their RMT counterparts, developed by Simons and Altshuler  (see also related works  on statistics of level curvatures, as well as early papers  on ensembles interpolating between GOE and GUE and  on an impressively accurate verification of many aspects of RMT in acoustic waves experiments). Those and related developments resulted in a considerable broadening of the list of random matrix ensembles relevant to physical applications and amenable to analytical treatment beyond the classical Wigner-Dyson list. In particular, ensembles of random matrices which are not invariant with respect to changes of the basis attracted considerable attention, e.g. Band Random Matrices , which allowed for the effects of the Anderson localization to be correctly incorporated at the level of RMT. Moreover, the Dyson three-fold symmetry classification of underlying Hamiltonians was discovered to be insufficient for describing spectra appearing in Quantum Chromodynamics  as well as in disordered superconducting structures and was replaced by a more comprehensive list . Around the same time systematic investigations of non-Hermitian deformations of random matrix ensembles were undertaken due to their relevance for description of statistics of S-matrix poles (resonances) in quantum chaotic scattering  and in Quantum Chromodynamics with non-zero chemical potential , which also lead to a general boost of interest in their properties .
On the other hand