Cognitive science applies diverse tools and perspectives to study human language. Recently, an exciting body of work has examined linguistic phenomena through the lens of efficiency in usage: what otherwise puzzling features of language find explanation in formal accounts of how language might be optimized for communication and learning? Here, we review studies that deploy formal tools from probability and information theory to understand how and why language works the way that it does, focusing on phenomena ranging from the lexicon through syntax. These studies show how a pervasive pressure for efficiency guides the forms of natural language and indicate that a rich future for language research lies in connecting linguistics to cognitive psychology and mathematical theories of communication and inference.