Human language has unusual structural properties that enable open-ended communication. In recent years, researchers have begun to appeal to cultural evolution to explain the emergence of these structural properties. A particularly fruitful approach to this kind of explanation has been the use of laboratory experiments. These typically involve participants learning and interacting using artificially constructed communication systems. By observing the evolution of these systems in the lab, researchers have been able to build a bridge between individual cognition and population-wide emergent structure. We review these advances, and show how cultural evolution has been used to explain the origins of structure in linguistic signals, and in the mapping between signals and meanings.