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The representation of physical and complex biological systems in terms of organismic supercategories was introduced in 1968 by Baianu and Marinescu in the attached paper which was published in the Bulletin of Mathematical Biophysics, edited by Nicolas Rashevsky. The different approaches to relational biology, developed by Rashevsky, Rosen and by Baianu et al.(1968,1969,1973,1974,1987,2004)were later discussed. The present paper is an attempt to outline an abstract unitary theory of systems. In the introduction some of the previous abstract representations of systems are discussed. Also a possible connection of abstract representations of systems with a general theory of measure is proposed. Then follow some necessary definitions and authors' proposals for an axiomatic theory of systems. Finally some concrete examples are analyzed in the light of the proposed theory. An abstract representation of biological systems from the standpoint of the theory of supercategories is presented. The relevance of such representations forG-relational biologies is suggested. In section A the basic concepts of our representation, that is class, system, supercategory and measure are introduced. Section B is concerned with the mathematical representation starting with some axioms and principles which are natural extensions of the current abstract representations in biology. Likewise, some extensions of the principle of adequate design are introduced in section C. Two theorems which present the connection between categories and supercategories are proved. Two other theorems concerning the dynamical behavior of biological and biophysical systems are derived on the basis of the previous considerations. Section D is devoted to a general study of oscillatory behavior in enzymic systems, some general quantitative relations being derived from our representation. Finally, the relevance of these results for a quantum theoretic approach to biology is discussed. http://www.springerlink.com/content/141l35843506596h/