Algebras of bounded operators are familiar, either as C^*-algebras or as von Neumann algebras. A first generalization is the notion of algebras of unbounded operators (O^*-algebras), mostly developed by the Leipzig school and in Japan (for a review, we refer to the monographs of K. Schmudgen 1990] and A. Inoue 1998]). This volume goes one step further, by considering systematically partial ^*-algebras of unbounded operators (partial O^*-algebras) and the underlying algebraic structure, namely, partial ^*-algebras. It is the first textbook on...