This chapter reviews competing analyses of depictive secondary predicate constructions such as the cat ate the mouse alive in the light of cross-linguistic data. Existing proposals for a syntactic relationship have in common that they aim to capture the semantic relationship between the depictive constituent (alive) and one of the arguments of the main predicate (in this case, the mouse) by variously representing the depictive as a direct (secondary) predicate, as the predicate in a small clause, or in a general relationship of syntactic dependency with this argument (the controller). Evidence for such a relationship comes from languages that show agreement between depictives and their controllers. However, the expression types singled out by agreement exhibit a great degree of cross-linguistic variation and include expressions traditionally classified as adverbials (i.e. as event-oriented rather than argument-oriented), raising the question of whether a predication relationship can be maintained. Other issues discussed include syntactic and semantic constraints on controllers, and the delimitation of depictives from predicative complements and complex predicates. The chapter concludes with a critical review of proposed extensions of the notion of secondary predicates to account for the syntactic freedom of so-called non-configurational languages and for possessor-raising constructions.