This survey article by Vilmos Totik provides an overview of the theory of orthogonal polynomials on the real line, excluding those on the unit circle. The article is divided into several sections, covering various aspects of the theory, including: 1. **Introduction**: The author introduces the two main parts of the theory: algebraic and analytical aspects. The algebraic part deals with special functions, combinatorics, and algebra, while the analytical part focuses on questions typical in mathematical analysis. 2. **Orthogonal Polynomials with Respect to Measures**: The article explains how to generate orthogonal polynomials with respect to a positive Borel measure on the complex plane. It also discusses the Riemann-Hilbert approach, which is a powerful method for constructing these polynomials. 3. **Orthogonal Polynomials with Respect to Inner Products**: The article explores the concept of orthogonal polynomials with respect to inner products and the conditions under which they can be explicitly constructed. 4. **Varying Weights**: The behavior of orthogonal polynomials with respect to varying measures is discussed, with examples such as Freud weights and multipoint Padé approximation. 5. **Matrix Orthogonal Polynomials**: The article highlights the rich theory of matrix orthogonal polynomials, which is more complex than the scalar case. 6. **The \(L^2\) Extremal Problem**: The extremal problem of minimizing the \(L^2\) norm for monic polynomials is discussed, leading to the Christoffel functions and their properties. 7. **Classical Orthogonal Polynomials**: The article covers Jacobi, Laguerre, and Hermite polynomials, which are special cases with additional properties such as derivatives that are also orthogonal polynomials. 8. **Padé Approximation and Rational Interpolation**: The article explains how orthogonal polynomials are used in Padé approximation and rational interpolation, particularly in the context of Markov functions. 9. **Moment Problem**: The moment problem is discussed, including the existence and uniqueness of measures given a sequence of moments. 10. **Jacobi Matrices and Spectral Theory**: The spectral theory of self-adjoint operators associated with Jacobi matrices is explored. 11. **Quadrature**: The article discusses quadrature rules and their accuracy, emphasizing the role of orthogonal polynomials in optimal quadrature. 12. **Random Matrices**: The connection between random matrices and orthogonal polynomials is explained, including the density of states and the behavior of eigenvalues. 13. **Heuristics**: The article provides heuristics on the behavior of orthogonal polynomials, including their zeros, asymptotics, and universal properties. The article is intended for non-experts and includes introductory materials, making it accessible to a wide audience. It also references several recent monographs and books for further reading.This survey article by Vilmos Totik provides an overview of the theory of orthogonal polynomials on the real line, excluding those on the unit circle. The article is divided into several sections, covering various aspects of the theory, including: 1. **Introduction**: The author introduces the two main parts of the theory: algebraic and analytical aspects. The algebraic part deals with special functions, combinatorics, and algebra, while the analytical part focuses on questions typical in mathematical analysis. 2. **Orthogonal Polynomials with Respect to Measures**: The article explains how to generate orthogonal polynomials with respect to a positive Borel measure on the complex plane. It also discusses the Riemann-Hilbert approach, which is a powerful method for constructing these polynomials. 3. **Orthogonal Polynomials with Respect to Inner Products**: The article explores the concept of orthogonal polynomials with respect to inner products and the conditions under which they can be explicitly constructed. 4. **Varying Weights**: The behavior of orthogonal polynomials with respect to varying measures is discussed, with examples such as Freud weights and multipoint Padé approximation. 5. **Matrix Orthogonal Polynomials**: The article highlights the rich theory of matrix orthogonal polynomials, which is more complex than the scalar case. 6. **The \(L^2\) Extremal Problem**: The extremal problem of minimizing the \(L^2\) norm for monic polynomials is discussed, leading to the Christoffel functions and their properties. 7. **Classical Orthogonal Polynomials**: The article covers Jacobi, Laguerre, and Hermite polynomials, which are special cases with additional properties such as derivatives that are also orthogonal polynomials. 8. **Padé Approximation and Rational Interpolation**: The article explains how orthogonal polynomials are used in Padé approximation and rational interpolation, particularly in the context of Markov functions. 9. **Moment Problem**: The moment problem is discussed, including the existence and uniqueness of measures given a sequence of moments. 10. **Jacobi Matrices and Spectral Theory**: The spectral theory of self-adjoint operators associated with Jacobi matrices is explored. 11. **Quadrature**: The article discusses quadrature rules and their accuracy, emphasizing the role of orthogonal polynomials in optimal quadrature. 12. **Random Matrices**: The connection between random matrices and orthogonal polynomials is explained, including the density of states and the behavior of eigenvalues. 13. **Heuristics**: The article provides heuristics on the behavior of orthogonal polynomials, including their zeros, asymptotics, and universal properties. The article is intended for non-experts and includes introductory materials, making it accessible to a wide audience. It also references several recent monographs and books for further reading.