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. Schmüdgen [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 this topic. The first part is devoted to partial O-algebras, basic properties, examples, topologies on them. The climax is the generalization to this new framework of the celebrated modular theory of Tomita-Takesaki, one of the cornerstones for the applications to statistical physics. The second part focuses on abstract partial -algebras and their representation theory, obtaining again generalizations of familiar theorems (Radon-Nikodym, Lebesgue).