 as the Circular Orthogonal Ensemble (COE), the Circular Unitary Ensemble (CUE) and the Circular Symplectic Ensemble (CSE). Finally deserves mentioning the so-called Ginibre Ensemble of matrices with independent, identically and normally distributed real, complex, or quaternion real entries, and no further constraints imposed. The corresponding eigenvalues are scattered in the complex plane.
Brief History and Background Information
Although origins of RMT could be traced back to works by Wishart (1928)  and James (1954-1964)  in the field of Statistics (see also Hua's works in multivariate harmonic analysis ) the real start of the field is usually attributed to highly influential papers by Eugene Wigner in 1950's  motivated by applications in Nuclear Physics. Wigner suggested that fluctuations in positions of compound nuclei resonances can be described in terms of statistical properties of eigenvalues of very large real symmetric matrices with independent, identically distributed entries. The rational behind such a proposal was the idea that in the situation when it is hardly possible to understand in detail individual spectra associated with any given nucleus composed of many strongly interacting quantum particles, it may be reasonable to look at the corresponding systems as "black boxes" and adopt a kind of statistical description, not unlike thermodynamics approach to classical matter. A system in Quantum Mechanics can be characterized by a self-adjoint linear operator in Hilbert space, its Hamiltonian, which we may think informally of as a matrix of infinitely many dimensions. This suggests that input to such theory should be very general properties of the underlying generic Hamiltonians, most importantly such global features as the Hermiticity, the time-inversion invariance as well as other symmetries Hamiltonians may obey from general principles. Wigner hoped that the output of the model will be universal, that is independent on the detail and common to majority of systems sharing the corresponding symmetries. 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 sub