This paper concern the study of *-algebras, that Is to say, algebras of operators, bounded or not, defined on a dense invariant domain X) of some Hilbert space H. Such objects, suitably topologized, are natural DF-spaces of analysis [5], and the aim of this paper is to present a development similar to that of C*-algebras. The work has been divided as follows. In first part, we examine properties of the domain X), and this leads to the analysis of the set B(!D, T>) of continuous sesquilinear forms on T> x X): this space is a DF-space, and admits a predual which is a Frechet space. Of course, B(T>, T>) plays the role of the algebra of all bounded operators of C*-algebras. The question of normality of the positive cone of $1 is solved for particular *-algebras 31 (see part three), and its central role is described in Proposition 6. Second part describes the second dual of 21. Third part is concerned with particular *algebras, for which we get an explicit description of the dual. The topological contents of such algebras are quite opposite to that of C*-theory. It should be pointed out that the study of positive linear forms on special *-algebras has been undertaken, without topology, by Shermann [11] and Woronowicz [12], and their methods are connected to our topological analysis. Theory of duality in locally convex spaces, mainly in Frechet spaces, is essential for our study, and our bibliography in this direction is very incomplete. The reader will find main notations of this work in [9]: though [8] contains [9], its knowledge is not necessary for the understanding of this paper.