This book presents the theory of fractional calculus formulated by an operator-based method. The differentiation operator in the classical calculus is extended to a general linear operator which satisfies Leibniz's rule. The key idea in this theory is to consider a sequence of functions produced by iterated applications of such a linear operator to a function which belongs to some specified function space. The fractional power of such a linear operator is defined by extending the integral representation of a function in this sequence by analytic continuation. This book covers various contents of the fractional calculus