Comm. in Alg. 19 (1991), 379–417. DERIVED CATEGORIES AND UNIVERSAL PROBLEMS Bernhard Keller Mathematik, G 28.2 ETH-Zentrum 8092 Zurich, Switzerland Introduction In this paper we search for a universal property of the (bounded positive) derived category of an exact category. We thereby hope to obtain a better understanding of the category of S -functors [10] starting from the derived category. Such S -functors play an essential rˆole in the study of hearts of t- structures [2] [1], in J. Rickard’s ’Morita theory for derived categories‘ [13] and in D. Happel’s description of the derived category of a finite-dimensional algebra [7]. We briefly outline the contents of the paper. Let A be an exact category and DA the bounded positive derived category (cf. section 1). We start with what we consider the most natural approach, namely the question whether the canonical ∂ -functor A→DA is universal among the ∂ -functors from A to suspended categories S . This, however, is not the case. We analyse the situation in section 1. Our conclusion is that the concepts of suspended category and S -functor alone do not provide rich enough a framework for an adequate treatment of the question. As a supplement, we propose ’towers‘ of