The book is devoted to the generalization of operations of differentiation and integration of functions of one and many variables from whole orders to fractional ones. Actual and complex , as well as applications of fractional integration and differentiation theory to integral and differential equations , Theory of Functions . In it, for the first time in the world monographic literature, the classical and modern results of this theory are systematically presented. At the end of each chapter, historical information and reviews of works on the topic of the chapter . are given. The book is encyclopedic in nature, covers a wide variety of known forms of fractional integration and differentiation and a huge number of publications in this area until 1986. The inclusion of the necessary preliminary information in the book and a detailed presentation of the evidence make it available to students of physical and mathematical faculties, as well as technical universities. Designed for mathematicians, physicists, mechanics, engineers, teachers, graduate students and students interested in mathematical analysis and its applications.