Traditional accounts of language postulate two basic components: words stored in a lexicon, and rules that govern how they can be combined into meaningful sentences, a grammar. But, although this words-and-rules framework has proven itself to be useful in natural language processing and cognitive science, it has also shown important shortcomings when faced with actual language use. In this article, we review evidence from language acquisition, sentence processing, and computational modeling that shows how multiword expressions such as idioms, collocations, and other meaningful and common units that comprise more than one word play a key role in the organization of our linguistic knowledge. Importantly, multiword expressions straddle the line between lexicon and grammar, calling into question how useful this distinction is as a foundation for our understanding of language. Nonetheless, finding a replacement for the foundational role the words-and-rules approach has played in our theories is not straightforward. Thus, the second part of our article reviews and synthesizes the diverse approaches that have attempted to account for the central role of multiword expressions in language representation, acquisition, and processing.