In this paper, we look in detail at some of the particular micro-processes involved in the online, relevance-driven derivation of explicature and implicatures. But before that, we set out some background intellectual history tracing the development of the concept of implicature. It did not arise within linguistics or cognitive science but within the philosophy of language, where its main purpose was to help in the delineation of a favoured notion of 'semantic content'. Its subsequent adoption into the cognitively-based relevance-theoretic account of communication where it is placed in opposition to explicature rather than semantics has naturally led to its being somewhat altered with regard to the domain of utterance meaning it encompasses and the role it plays.