Free Probability and Random Matrices 1. Asymptotic Freeness of Gaussian Random Matrices 2. The Free Central Limit Theorem and Free Cumulants 3. Free Harmonic Analysis 4. Asymptotic Freeness 5. Second Order Freeness 6. Free Group Factors and Freeness 7. Free Entropy X the Microstates Approach via Large Deviations Free Entropy X* the Non Microstates Approach via Free Fisher Information 9. Operator Valued Free Probability Theory and Block Random Matrices 10. Polynomials in Free Variables and Operator Valued Convolution 11. Brown Measure Solutions to Exercises References Index of Exercises This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras free probability has intertwined with non crossing partitions random matrices applications in wireless communications representation theory of large groups quantum groups the invariant subspace problem large deviations subfactors and beyond. This book puts a special emphasis on the relation of free probability to random matrices but also touches upon the operator algebraic combinatorial and analytic aspects of the theory. The book serves as a combination textbook research monograph with self contained chapters exercises scattered throughout the text and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras combinatorics analytic functions or applications in engineering and statistical physics