Abstract: We investigate the (unbounded) derived category of a differential Z-graded category (=DG category).As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch.V, characterizing module categories among abelian categories.After adapting some homological algebra we go on to prove a "Morita theorem" (8.2) generalizing results of [19] and [20].Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. SummaryWe give an account of the contents of this paper for the special case of DG algebras.Let k be a commutative ring and A a DG (k-)algebra, i.e. a Z-graded fe-algebra A=®Ap eZ endowed with a differential d of degree 1 such that d(ab)=(da)b-^(-l) p a(db)for all ae^, be A. A DG (right) A-module is a Z-graded A-module M=@M P peZ endowed with a differential d of degree 1 such thatfor all m e M^, a e A. A morphism of DG A-modules is a homogeneous morphism of degree 0 of the underlying graded A-modules commuting with the differentials.The DG A-modules form an abelian category ^ A. A morphism /: M -> N of ^ A is nullhomotopic if/= dr-\-rd for some homogeneous morphism r: M -> N of degree -1 of the underlying graded A-modules.The homotopy category ^ A has the same objects as ^ A. Its morphisms are residue classes of morphisms of ^A modulo null-homotopic morphisms.It is a triangulated [23] category (2.2).A quasi-isomorphism is a morphism of ^A inducing isomorphisms in homology.