This book is devoted to the new method of integrating physically important nonlinear equations that is based on the replacement of the original equation by an equation of the same form with respect to unknown functions assuming values in a suitable operator algebra. The generality of this new method can be characterized with its applicability to investigation of the Korteweg-de Vries, Kadomtsev-Petviashvili, nonlinear Schrödinger and sine-Gordon equations, Toda and Langmuir chains and so on. The book contains 4 chapters: General Scheme; Realization of the General Scheme in Matrix Rings and N-Soliton Solutions; Realization of General Scheme in Operator Algebras; Classes of Solutions to Nonlinear Equations. The book is of interest to mathematicians and physists as well as lectures, post-graduate and graduate students at universities.