We examine the evolving concept of what constitutes a non-native (or alien) freshwater fish. In an attempt to distinguish between biogeographical and socio-political perspectives, we review the patterns in the introduction and dispersal of non-native fishes in Europe and North America, and especially the recent expansion of Ponto-Caspian gobies in Europe. We assess patterns in the development of national policy and legislation in response to the perceived threat of non-native fish introductions to native species and ecosystems. We review, and provide a glossary of, the terms and definitions associated with non-native species. Finally, we discuss perspectives as regards the future treatment of naturalized species.