Semantic theories of cumulativity vary in several respects, including (i) whether cumulativity is limited to lexical predicates and (ii) whether there are cumulation operators in the object language. We address the cross-linguistic predictions of different settings of these two parameters and evaluate them in light of a preliminary set of data from 22 languages, largely collected from native-speaker linguists. We submit that cumulative readings of non-lexical predicates are available cross-linguistically. We then address the question whether there are overt morphemes that behave like the cumulation operators **, ***, etc. Our data only give a partial answer, since there are different ways of integrating such operators into the grammar. No language in our sample had overt markers that were required for a cumulative reading, but absent in case of a distributive reading. Assuming that the LFs of distributive readings do not have to contain such cumulation operators, our data set does not provide evidence for their existence.