In this article, I discuss Ancient Greek constructions consisting of a form of the verb eimi ‘I am’ and a present, perfect or aorist participle. In particular, I focus on those uses where the participle is said to have an “adjectival” function. My main goal is to provide a unified semantic description of this phenomenon, adopting a cognitive framework. I show that adjectival periphrasis typically involves the predication of properties, which can be characterized in terms of low transitivity (Hopper & Thompson, 1980). I furthermore argue that a so-called “property reading” involves a particular kind of conceptual integration, whereby only one component state of the verb eimi is elaborated by the participle.