The ongoing development and analysis of efficient algorithms for solving positive definite Toeplitz equations is motivated to a large extent by the importance of these equations in signal processing applications. The role of positive definite Toeplitz matrices in this and other areas of mathematics and engineering stems from Schur's study of bounded analytic functions on the unit disk, and Szegő's theory of polynomials orthogonal on the unit circle. These ideas underlie several Toeplitz solvers, and provide a useful framework for understanding the relationships among these algorithms. In this paper we give an overview of several direct algorithms for solving positive definite Toeplitz systems of linear equations from this classical viewpoint.