This paper provides a review of Toeplitz and circulant matrices, focusing on their asymptotic behavior, eigenvalues, and applications in stochastic time series analysis. The author, Robert M. Gray, presents a tutorial overview of these matrices, emphasizing conceptual simplicity and intuitive understanding over mathematical complexity. The text is structured into chapters covering the asymptotic behavior of matrices, circulant matrices, Toeplitz matrices, matrix operations on these matrices, and their applications in stochastic time series. Toeplitz matrices are defined by constant diagonals, while circulant matrices are formed by cyclic shifts of their rows. The paper discusses the eigenvalues, eigenvectors, and matrix operations of both types, highlighting their asymptotic properties. Key results include Szegö's theorem, which describes the asymptotic distribution of eigenvalues of Toeplitz matrices, and the asymptotic equivalence of sequences of matrices. The paper also explores applications of these matrices in signal processing, particularly in the analysis of Gaussian processes, the rate-distortion function of Gaussian processes, and one-step prediction error. It emphasizes the use of Toeplitz and circulant matrices in modeling discrete-time random processes and their covariance structures. The author provides a detailed treatment of matrix norms, eigenvalue distributions, and asymptotic equivalence, demonstrating how these concepts can be applied to evaluate limits of solutions to finite-dimensional problems. The paper concludes with a discussion of the broader implications of these results in various fields, including information theory, coding, spectral estimation, and signal processing. The review aims to make these advanced mathematical concepts accessible to engineers and applied mathematicians who may lack the background in pure mathematics.This paper provides a review of Toeplitz and circulant matrices, focusing on their asymptotic behavior, eigenvalues, and applications in stochastic time series analysis. The author, Robert M. Gray, presents a tutorial overview of these matrices, emphasizing conceptual simplicity and intuitive understanding over mathematical complexity. The text is structured into chapters covering the asymptotic behavior of matrices, circulant matrices, Toeplitz matrices, matrix operations on these matrices, and their applications in stochastic time series. Toeplitz matrices are defined by constant diagonals, while circulant matrices are formed by cyclic shifts of their rows. The paper discusses the eigenvalues, eigenvectors, and matrix operations of both types, highlighting their asymptotic properties. Key results include Szegö's theorem, which describes the asymptotic distribution of eigenvalues of Toeplitz matrices, and the asymptotic equivalence of sequences of matrices. The paper also explores applications of these matrices in signal processing, particularly in the analysis of Gaussian processes, the rate-distortion function of Gaussian processes, and one-step prediction error. It emphasizes the use of Toeplitz and circulant matrices in modeling discrete-time random processes and their covariance structures. The author provides a detailed treatment of matrix norms, eigenvalue distributions, and asymptotic equivalence, demonstrating how these concepts can be applied to evaluate limits of solutions to finite-dimensional problems. The paper concludes with a discussion of the broader implications of these results in various fields, including information theory, coding, spectral estimation, and signal processing. The review aims to make these advanced mathematical concepts accessible to engineers and applied mathematicians who may lack the background in pure mathematics.