Winner of the 2015 Bolyai Prize of the Hungarian Academy of Sciences! This twopart volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of onedimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by \(z\) (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. The book is suitable for graduate students and researchers interested in analysis. Readership Graduate students and research mathematicians interested in analysis. Reviews "Simon's work is not just a book about orthogonal polynomials but also about probability measures on onedimensional Schrödinger operators and operator theory. It is extremely complex, multilayered, fascinating, and inspiring, while remaining very readable (even for advanced students). Without a doubt this monograph will become the standard reference for the theory of orthogonal polynomials on the unit circle for a long time to come."  Jahresbericht der DMV "Undoubtedly that ... this book will become a standard reference in the field tracing the way for future investigations on orthogonal polynomials and their applications. Combining methods from various areas of analysis (calculus, real analysis, functional analysis, complex analysis) as well as by the importance of the orthogonal pholynomials in applications, the book will have a large audience including researchers in mathematics, physics, (and) engineering."  Stefan Cobzas, Studia Universitatis BabesBolyai, Mathematica Table of Contents  Part 1: The Basics
 Szegő's theorem
 Tools for Geronimus' theorem
 Matrix representations
 Baxter's theorem
 The strong Szegő theorem
 Verblunsky coefficients with rapid decay
 The density of zeros
 Bibliography
 Author index
 Subject index
 Part 2: Rakhmanov's theorem and related issues
 Techniques of spectral analysis
 Periodic Verblunsky coefficients
 Spectral analysis of specific classes of Verblunsky coefficients
 The connection to Jacobi matrices
 Appendix A. Reader's guide: Topics and formulae
 Appendix B. Perspectives
 Appendix C. Twelve great papers
 Appendix D. Conjectures and open questions
 Bibliography
 Author index
 Subject index
