This paper, the first of two, is devoted to a systematic study of partial *-algebras of closable operators in a Hilbert space (partial Op*-algebras). After setting up the basic definitions, we describe canonical extensions of partial Op*-algebras by closure and introduce a new bounded commutant, called quasi-weak. We initiate a theory of abelian partial *-algebras.As an application, we analyze thoroughly the partial Op*-algebras generated by a single closed symmetric operator. §1. Introduction Ever since the pioneering days of Heisenberg's Matrix Mechanics, operator algebras have played a prominent role in quantum theories. For instance, the algebraic language is by now standard in quantum statistical mechanics (e. g. see the monograph of Bratteli and Robinson [1]). However the algebras used in this context consist invariably of bounded operators (in particular representations of abstract C*-algebras) and their bicommutants, that is, von Neumann algebras. In particular the latter play a crucial role in the Tomita-Takesaki theory [1].Yet this framework is often too narrow for applications. The next step is to consider algebras of unbounded operators, consisting of operators with a common dense invariant domain. Take for instance a nonrelativistic one particle quantum system. In the Schrodinger representation, with Hilbert space L 2 (R 3 ), the canonical variables are represented by the operators q and p, both unbounded and obeying the canonical commutation relations [p j? q k~] = id jk . The natural domain associated to this system is of course Schwartz space ^(R 3 ), and the corresponding *-algebra ^(£f) consists of all operators A such that A£f c £f and A*tf a