This book is unique. It contains a detailed account of all important relations in the analytic theory of determinants from the classical work of Laplace, Cauchy and Jacobi in the 18th and 19th centuries to the most recent 20th century developments. Several contributions have never been published before. The first five chapters are purely mathematical in nature and make extensive use of the column vector notation and scaled cofactors. They contain a number of important relations involving derivatives which prove beyond a doubt that the theory of determinants has emerged from the confines of classical algebra into the brighter world of analysis. The whole of Chapter 4 is devoted to particular determinants including alternants, Wronskians and Hankelians. The contents of Chapter 5 include the Cusick and Matsuno identities. Chapter 6 is devoted to the verifications of the known determinantal solutions of several nonlinear equations which arise in three branches of mathematical physics, namely lattice, soliton and relativity theory. They include the KdV, Toda and Einstein equations. The solutions are verified by applying theorems established in earlier chapters and in the extensive appendix. The book ends with an extensive bibliography and an index. Mathematicians, physicists and engineers who wish to become acquainted with modern developments in the analytic theory of determinants will find the book indispensable.