The chapter provides an overview of Toeplitz and Circulant Matrices, focusing on their asymptotic behavior, eigenvalues, norms, and applications. Toeplitz matrices are characterized by their Toeplitz structure, where each row is a right cyclic shift of the previous row, while Circulant matrices are a special case of Toeplitz matrices with each row being a rotation of the first row. The chapter discusses the asymptotic behavior of eigenvalues, inverses, and products of these matrices, emphasizing the use of asymptotic equivalence to simplify proofs. Key theorems include Szegö's theorem on the asymptotic eigenvalue distribution of Hermitian Toeplitz matrices, which relates the eigenvalues to the power spectral density of a stationary process. The chapter also covers matrix norms, asymptotically equivalent sequences of matrices, and asymptotically absolutely equal distributions, providing rigorous proofs and examples to illustrate the concepts. Applications to signal processing, such as the differential entropy rate of a Gaussian process and the rate-distortion function, are discussed to highlight the practical significance of these theoretical results.The chapter provides an overview of Toeplitz and Circulant Matrices, focusing on their asymptotic behavior, eigenvalues, norms, and applications. Toeplitz matrices are characterized by their Toeplitz structure, where each row is a right cyclic shift of the previous row, while Circulant matrices are a special case of Toeplitz matrices with each row being a rotation of the first row. The chapter discusses the asymptotic behavior of eigenvalues, inverses, and products of these matrices, emphasizing the use of asymptotic equivalence to simplify proofs. Key theorems include Szegö's theorem on the asymptotic eigenvalue distribution of Hermitian Toeplitz matrices, which relates the eigenvalues to the power spectral density of a stationary process. The chapter also covers matrix norms, asymptotically equivalent sequences of matrices, and asymptotically absolutely equal distributions, providing rigorous proofs and examples to illustrate the concepts. Applications to signal processing, such as the differential entropy rate of a Gaussian process and the rate-distortion function, are discussed to highlight the practical significance of these theoretical results.