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The terminology used in definitions of different types of fixed expressions (such as idioms, locutions, proverbs etc.) is complex and sometimes ambiguous. Within the field of phraseology, it is therefore subject to discussion. This chapter deals with the complexity of the notion of non-compositionality. Non-compositionality is a criterion that often appears in definitions of fixed expressions. It will be shown that there are four dichotomies of particular interest within the very notion of non-compositionality. These dichotomies are: motivation/ non-motivation, transparency/opacity, analysability/unanalyzability and literal/figurative meaning. Other notions of relevance to non-compositionality, such as common meaning, prototypicality, salience and frequency as well as encyclopedic compositionality will also be described and problematized. Finally, the notions of partial compositionality, holistic or atomistic compositionality, and lexical content or grammatical form, all describing different aspects of compositionality, will be evoked, to further show the complexity of this criterion of fixedness.