This text is devoted to the theory of entire operators, which was founded by M.G. Krein. The theory lies at the junction of the spectral theory of Hermitian operators and the theory of analytic functions, combining the methods of each. The purpose of this text is to show how problems of both classical and modern analysis can be looked at from the point of view of entire operator theory. This is one of the first systematic presentations of the basic concepts of Krein's theory and its applications, and its study of his unpublished lectures and his works gives recognition to the unique approach he developed, which was not widely known for several years. This book is suitable for postgraduate students as well as researchers interested in the spectral theory of operators, complex analysis, differential equations and extrapolation problems.