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In this monograph, we consider free calculus on operator algebras. In Part 1, we consider structure theorems for a C -algebra generated by a finite family F of partial isometries on a Hilbert space. We show that, whenever such a family is given, there always exists the corresponding graph G of F. Moreover, the graph groupoid of G shows how this family F works on H. We realize that C -algebra C (F), generated by F is -isomorphic to the groupoid C -algebra generated by the graph groupoid of G, under suitable (embedding) representations. In Part 2, we determine free calculus on C (F). Free calculus consists of the differentiation, and the integration, like the usual calculus. The fundamental properties of them are studied. Also, we investigate free calculus on arbitrary C -algebras. In Part 3, we extend the free calculus in the sense of Part 2 to that on groupoid C -dynamical systems.