Partial *-algebras, a tool for the mathematical description of physical systems ?

We review the main steps in the development of partial *-algebras. First we discuss the algebraic structure stemming from the partial multiplication. Then we study in some detail the locally convex partial *-algebras, in particular, the Banach partial *-algebras, and we describe a number of concrete examples. Next we consider the partial *-algebras of closable operators in Hilbert spaces (partial O*-algebras), with a special emphasis on their *-automorphisms. Finally we sketch the representation theory of abstract partial *-algebras and give some instances of possible physical applications