- First version received May 1990; final version accepted January 1993 (Eds.) This paper provides necessary and sufficient conditions for the existence of greatest and maximal elements of weak and strict preferences, and unifies two very different approaches used in the related literature (the convexity and acyclicity approaches). Conditions called transfer FS-convexity and transfer SS-convexity are shown to be necessary and, in conjunction with transfer closedness and transfer openness, sufficient for the existence of greatest and maximal elements of weak and strict preferences, respectively. The results require neither the continuity nor convexity of preferences, and are valid for both ordered and unordered binary relations. Thus, the results generalize almost all of the theorems on existence of maximal elements of preferences that appear in the literature. 1993 The Review of Economic Studies Limited.