This chapter provides an introduction to the theory of random matrices, focusing on the distribution of eigenvalues. It begins with numerical experiments to illustrate basic aspects of the subject, such as the Wigner semicircle law and level repulsion. The chapter then delves into the invariant measures for the three circular ensembles involving unitary matrices, defining level spacing distributions and expressing them in terms of Fredholm determinants. It discusses the modification of these measures for orthogonal polynomial ensembles and the universality of level spacing distribution functions in a scaling limit. The chapter also covers the discovery by Jimbo, Miwa, M\"ori, and Sato that the Fredholm determinant is a tau-function, presenting a simplified derivation of the nonlinear completely integrable equations they derived. The case of a single interval is discussed, where these equations reduce to a Painlevé V equation. Finally, the chapter addresses the asymptotics of level spacing distribution functions for large intervals, using both the Painlevé representation and new results in Toeplitz/Wiener-Hopf theory.This chapter provides an introduction to the theory of random matrices, focusing on the distribution of eigenvalues. It begins with numerical experiments to illustrate basic aspects of the subject, such as the Wigner semicircle law and level repulsion. The chapter then delves into the invariant measures for the three circular ensembles involving unitary matrices, defining level spacing distributions and expressing them in terms of Fredholm determinants. It discusses the modification of these measures for orthogonal polynomial ensembles and the universality of level spacing distribution functions in a scaling limit. The chapter also covers the discovery by Jimbo, Miwa, M\"ori, and Sato that the Fredholm determinant is a tau-function, presenting a simplified derivation of the nonlinear completely integrable equations they derived. The case of a single interval is discussed, where these equations reduce to a Painlevé V equation. Finally, the chapter addresses the asymptotics of level spacing distribution functions for large intervals, using both the Painlevé representation and new results in Toeplitz/Wiener-Hopf theory.