Introduction. M. Auslander has given, in unpublished notes, an axiomatic treatment of the classical primary decomposition theory for modules over commutative Noetherian rings. The purpose of this paper is to show that this axiom scheme may be suitably abstracted and modified so as to include also the recent Lesieur-Croisot theory [1 -3] of tertiary decomposition in modules over arbitrary (i.e., not necessarily commutative) Noetherian rings. We do this by developing an abstract theory of what we call decomposition functors, having both the primary and tertiary theories as special cases. Briefly, if A is any ring, a decomposition functor F assigns to a pair (M, N), consisting of a A-module M and submodule N, a collection F(M, N) of ideals in A. This assignment is subject to several axioms which, a posteriori at least, can be considered as embodying the essential features of the primary and tertiary decomposition theories. In ??1 and 2 we give the axioms for these decomposition functors and develop some of their basic properties. The material in these two sections is, apart from its more general content, a rearrangement and elaboration of the axioms given by Auslander. In ?3 we summmarize the basic facts concerning the primary and tertiary theories, and show how these may be defined within the framework of decomposition functors. Returning to the abstract theory, ?4 presents certain results concerning the comparison of two decomposition functors. In ?5 we define the concept of normal decomposition functors; the primary and tertiary theories are examples of such decompositions. Putting together the results of ??4 and 5 we obtain the following remarkable result: for finitely generated modules over a Noetherian ring, any two normal decomposition theories coincide, i.e., assign the same set F(M, N) of ideals to a pair (M, N). Using the fact that the tertiary theory is normal, we conclude that for finitely generated modules over a Noetherian ring, the tertiary theory is the unique normal decomposition theory. In this sense the tertiary decomposition theory is the only natural generalization, to arbitrary Noetherian rings, of the classical primary decomposition theory in commutative rings. This result answers a question of Goldie's [4, p. 127], and underlines the potential importance of the tertiary theory for the study of Noetherian rings.