This book explores generalized fractional calculus and its applications, covering a wide range of topics in fractional integration, differentiation, and related operators. It begins with the definition and properties of generalized fractional integration operators, including their integral transform representations and connections to Erdélyi-Kober operators. The text then discusses the properties of Riemann-Liouville and Weyl type generalized fractional integrals, as well as generalized fractional differentiation operators and their inversion formulas. It also examines the properties and examples of generalized Erdélyi-Kober derivatives. The book then moves on to recent developments in classical Erdélyi-Kober operators, including their convolutions, generalized differentiation and integration operators, and representations of their commutants. It introduces hyper-Bessel differential and integral operators, discussing their role as generalized fractional differintegrals, their projectors, and their applications in mathematical physics. The text also covers the transmutation method, including Poisson-Sonine-Dimovski transformations, and their applications to hyper-Bessel equations. The book further explores the Obrechkoff integral transform, its properties, inversion formulas, and applications. It then discusses applications to generalized hypergeometric functions, including Poisson-type integral representations and fractional derivative representations. The text concludes with further generalizations and applications of fractional calculus, including the use of Fox's H-function, solutions to integral equations, and generalized fractional calculus in analytic function classes. Appendices provide definitions, properties, and examples of special functions used throughout the book.This book explores generalized fractional calculus and its applications, covering a wide range of topics in fractional integration, differentiation, and related operators. It begins with the definition and properties of generalized fractional integration operators, including their integral transform representations and connections to Erdélyi-Kober operators. The text then discusses the properties of Riemann-Liouville and Weyl type generalized fractional integrals, as well as generalized fractional differentiation operators and their inversion formulas. It also examines the properties and examples of generalized Erdélyi-Kober derivatives. The book then moves on to recent developments in classical Erdélyi-Kober operators, including their convolutions, generalized differentiation and integration operators, and representations of their commutants. It introduces hyper-Bessel differential and integral operators, discussing their role as generalized fractional differintegrals, their projectors, and their applications in mathematical physics. The text also covers the transmutation method, including Poisson-Sonine-Dimovski transformations, and their applications to hyper-Bessel equations. The book further explores the Obrechkoff integral transform, its properties, inversion formulas, and applications. It then discusses applications to generalized hypergeometric functions, including Poisson-type integral representations and fractional derivative representations. The text concludes with further generalizations and applications of fractional calculus, including the use of Fox's H-function, solutions to integral equations, and generalized fractional calculus in analytic function classes. Appendices provide definitions, properties, and examples of special functions used throughout the book.