The theory of non-self-adjoint operators is necessary for the mathematical study of the processes that arise in non-conservative systems that play a large role in modern physics and mechanics. The book for the first time gives a detailed description of a number of methods of the theory of non-self-adjoint operators in Hilbert space (the method of assessing resolvents, the method of determining perturbations, various asymptotic methods, etc.) ). Along the way, new methods of obtaining various estimates, inequalities and ratios for eigenvalues and singular numbers of completely continuous operators are presented. Using these methods, a complete theory of symmetrically normalized ideals of completely continuous operators is given, in particular, such important ones as nuclear operators, Hilbert-Schmidt operators, etc. The material of the book can be used in university courses of linear algebra, integral equations and functional analysis. The book is addressed to researchers, graduate students and senior students - mathematicians, mechanics and theoretical physicists .