All existing types of fractional integro-differentiation are examined and compared. The application of fractional calculus to various types of equations is considered. These include first order integral equation (with power, power-logarithmic kernels and special functions in kernels), Euler-Poisson-Darboux-type equations, and differential equations of fractional order. The clear presentation of historical background, the extensive analysis of the great number of cited papers (more than 3000) and the authors' own significant research give this work the compactness of a handbook and the depth of an encyclopedia. This comprehensive monograph is devoted to the systematic and comprehensive exposition of classicial and modern results in the theory of fractional integrals and their applications. Various aspects of this theory, such as functions of one and several variables, periodical and non-periodical cases, and the technique of hypersingular integrals are studied.