Fractional calculus is a branch of mathematical analysis involved in the study of several different possibilities of defining real number powers or complex number powers of the differentiation operator, integration operator (D) and of the integration operator (J), and developing a calculus for such operators. It has several applications in multiple fields of engineering and science such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signal processing. Some important applications of fractional calculus include fractional conservation of mass, electrochemical analysis, groundwater flow problem, fractional advection dispersion equation, time-space fractional diffusion equation models, and structural damping models. This method of mathematical analysis is also utilized in PID controllers, acoustic wave equations for complex media, and fractional Schrödinger equation in quantum theory. This book outlines the theory and applications of fractional calculus in detail. Its extensive content provides the readers with a thorough understanding of the subject. The book, with its detailed analyses and data, will prove to be immensely beneficial for the professionals and students associated with the study of applied mathematics.