This paper provides an introduction to the theory of random matrices, focusing on the distribution of eigenvalues. The central quantity studied is $\tau(a) = \det(1 - K)$, where $K$ is an integral operator with kernel $\frac{1}{\pi}\frac{\sin\pi(x-y)}{x-y}\chi_{I}(y)$. In the Gaussian Unitary Ensemble (GUE), $\tau(a)$ represents the probability that no eigenvalues lie in the interval $I$. It is also a tau-function and is used to derive a system of nonlinear completely integrable equations, first introduced by Jimbo, Miwa, Möri, and Sato in 1980. For a single interval, these equations reduce to the Painlevé V equation. The paper discusses invariant measures for the three circular ensembles involving unitary matrices, level spacing distributions, and their universality in a particular scaling limit. It also explores the connection between the JMMS equations and integrable Hamiltonian systems. The asymptotic behavior of level spacing distribution functions for large intervals is analyzed, using the Painlevé representation and Toeplitz/Wiener-Hopf theory. Numerical experiments illustrate the distribution of eigenvalues in the Gaussian Orthogonal Ensemble (GOE) and the Wigner semicircle law. The paper also discusses the physical interpretation of the probability density $P_{N\beta}$, relating it to the 2D Coulomb potential for charges on a circle. The paper provides a simplified derivation of the JMMS equations, building on previous work by Mehta, Its, Izergin, Korepin, and Slavnov. It also discusses the reduction of the JMMS equations to the Painlevé V equation in the case of a single interval. The paper concludes with an analysis of the asymptotics of the level spacing distribution functions for large intervals, using the Painlevé V representation and other methods. The results are compared with continuum model calculations of Dyson. The paper provides a comprehensive overview of the theory of random matrices, focusing on the distribution of eigenvalues, invariant measures, level spacing distributions, and the connection to integrable systems.This paper provides an introduction to the theory of random matrices, focusing on the distribution of eigenvalues. The central quantity studied is $\tau(a) = \det(1 - K)$, where $K$ is an integral operator with kernel $\frac{1}{\pi}\frac{\sin\pi(x-y)}{x-y}\chi_{I}(y)$. In the Gaussian Unitary Ensemble (GUE), $\tau(a)$ represents the probability that no eigenvalues lie in the interval $I$. It is also a tau-function and is used to derive a system of nonlinear completely integrable equations, first introduced by Jimbo, Miwa, Möri, and Sato in 1980. For a single interval, these equations reduce to the Painlevé V equation. The paper discusses invariant measures for the three circular ensembles involving unitary matrices, level spacing distributions, and their universality in a particular scaling limit. It also explores the connection between the JMMS equations and integrable Hamiltonian systems. The asymptotic behavior of level spacing distribution functions for large intervals is analyzed, using the Painlevé representation and Toeplitz/Wiener-Hopf theory. Numerical experiments illustrate the distribution of eigenvalues in the Gaussian Orthogonal Ensemble (GOE) and the Wigner semicircle law. The paper also discusses the physical interpretation of the probability density $P_{N\beta}$, relating it to the 2D Coulomb potential for charges on a circle. The paper provides a simplified derivation of the JMMS equations, building on previous work by Mehta, Its, Izergin, Korepin, and Slavnov. It also discusses the reduction of the JMMS equations to the Painlevé V equation in the case of a single interval. The paper concludes with an analysis of the asymptotics of the level spacing distribution functions for large intervals, using the Painlevé V representation and other methods. The results are compared with continuum model calculations of Dyson. The paper provides a comprehensive overview of the theory of random matrices, focusing on the distribution of eigenvalues, invariant measures, level spacing distributions, and the connection to integrable systems.