The population genealogical processes associated with a wide range of exchangeable reproductive models (including the Wright-Fisher model) are shown to converge weakly, as the population size becomes large, to a particularly tractable limiting process, the age ordered analogue of Kingman's coalescent. This result extends the known convergence results for sample processes and effectively completes the robustness theory for neutral genealogies. Its consequences, which include a unification of the results for neutral models, have already been exploited elsewhere. The techniques used rely heavily on knowledge of sample behaviour, together with consistency arguments. They may be of more general interest.

Key Words: coupling, sampling functions, neutral models, exchangeability, genealogy 

AMS 1980 subject classifications: Primary 60F99, Secondary 60J75, 92A10