Contemporary proposals about universals are descriptive hypotheses about (a) the relation between translations and source texts (the equivalence relation), and (b) the relation between translations and comparable non-translations in the target language (the relation of textual fit). The article analyses the main trends in the thinking about translation universals, pointing out connections between the prescriptive tradition, work in literary translation criticism, and corpus-based research on universals. In order to characterize some of the main similarities and differences between these different approaches to universals, the author uses a classification of different kinds of hypotheses. After introducing these central notions, he places descriptive research in its historical context and then considers some of the terminological, conceptual and methodological problems of this kind of research.