1 Introduction From its inception,random matrix theory has been heavily influenced by its applications in physics,statistics and engineering.The landmark contributions to the theory of random matrices of Wishart(1928)[311], Wigner (1955)[303],and Marcenko and Pastur (1967)[170]were moti- vated to a large extent by practical experimental problems.Nowadays, random matrices find applications in fields as diverse as the Riemann hypothesis,stochastic differential equations,condensed matter physics, statistical physics,chaotic systems,numerical linear algebra,neural networks,multivariate statistics,information theory,signal processing, and small-world networks.Despite the widespread applicability of the tools and results in random matrix theory,there is no tutorial reference that gives an accessible overview of the classical theory as well as the recent results,many of which have been obtained under the umbrella of free probability theory. In the last few years,a considerable body of work has emerged in the communications and information theory literature on the fundamental limits of communication channels that makes substantial use of results in random matrix theory. The purpose of this monograph is to give a tutorial overview of ran- 31 Introduction From its inception, random matrix theory has been heavily influenced by its applications in physics, statistics and engineering. The landmark contributions to the theory of random matrices of Wishart (1928) [311], Wigner (1955) [303], and Mar˘cenko and Pastur (1967) [170] were moti￾vated to a large extent by practical experimental problems. Nowadays, random matrices find applications in fields as diverse as the Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing, and small-world networks. Despite the widespread applicability of the tools and results in random matrix theory, there is no tutorial reference that gives an accessible overview of the classical theory as well as the recent results, many of which have been obtained under the umbrella of free probability theory. In the last few years, a considerable body of work has emerged in the communications and information theory literature on the fundamental limits of communication channels that makes substantial use of results in random matrix theory. The purpose of this monograph is to give a tutorial overview of ran- 3