Conferences such as this and its predecessor in Périgueux present data on a scale that can often seem overwhelming, particularly to social scientists not versed in the complexities of population genetics and vice versa. The past has been complex in the extreme, and to recover all this complexity event by event would be impossible.We need to propose overarching hypotheses that can account for the comparative data from linguistics, genetics, and archaeology with as little stress as possible. Many of these hypotheses will inevitably include major component sof migration as well as interaction. The migrations discussed in the Geneva conference are significant examples of past expansions that have left indelible traces in the modern world. We need to analyse them if we are to understand our history.