The largest movements and replacements of human populations since the end of the Ice Ages resulted from the geographically uneven rise of food production around the world. The first farming societies thereby gained great advantages over hunter-gatherer societies. But most of those resulting shifts of populations and languages are complex, controversial, or both. We discuss the main complications and specific examples involving 15 language families. Further progress will depend on interdisciplinary research that combines archaeology, crop and livestock studies, physical anthropology, genetics, and linguistics.