The subject of this book is the fractional calculus which treats differentiation and integration to fractional order. This book presents a new approach to the fractional calculus being based on linear differentiator-like operators to arbitrary order. In this book, many theorems, corollaries, and lemmas are derived to reveal the properties of fractional differentiation and integration. Various special functions of arbitrary order are derived from this theory. Applications are also given to the solutions of basic differintegral equations, diffusion equations, Schroedinger's equations, electric circuits, and electromagnetic wave propagation problems.