This chapter introduces the field of random matrix theory and its applications, highlighting the historical origins and modern developments. It emphasizes the importance of random matrices in computational mathematics, statistics, and various scientific fields. The chapter outlines the basic questions in random matrix theory, such as the expectation of extreme eigenvalues and the probability of large deviations. It also discusses the role of random matrices as independent sums and provides an example of the sample covariance estimator to illustrate the application of matrix concentration inequalities. The chapter concludes with a discussion of the "arsenal of results" in matrix concentration, including matrix Gaussian and Rademacher series, Chernoff bounds, Bernstein bounds, and intrinsic dimension bounds. It also mentions other tools and recent developments, such as the method of exchangeable pairs and modified logarithmic Sobolev inequalities. The monograph aims to provide a comprehensive guide for graduate students and researchers, covering background material, key results, and practical applications.This chapter introduces the field of random matrix theory and its applications, highlighting the historical origins and modern developments. It emphasizes the importance of random matrices in computational mathematics, statistics, and various scientific fields. The chapter outlines the basic questions in random matrix theory, such as the expectation of extreme eigenvalues and the probability of large deviations. It also discusses the role of random matrices as independent sums and provides an example of the sample covariance estimator to illustrate the application of matrix concentration inequalities. The chapter concludes with a discussion of the "arsenal of results" in matrix concentration, including matrix Gaussian and Rademacher series, Chernoff bounds, Bernstein bounds, and intrinsic dimension bounds. It also mentions other tools and recent developments, such as the method of exchangeable pairs and modified logarithmic Sobolev inequalities. The monograph aims to provide a comprehensive guide for graduate students and researchers, covering background material, key results, and practical applications.