Abstract: Biological invasions are a continuous feature of a non equilibrium world, ever more so as a result of accidental and deliberate introductions by mankind. While many of these introductions are apparently harmless, others have significant consequences for organisms native to the invaded range, and entire communities may be affected. Here we provide a survey of common models of range expansion, and outline the consequences these models have for patterns in genetic diversity and population structure. We describe how patterns of genetic diversity at a range of markers can be used to infer invasion routes, and to reveal the roles of selection and drift in shaping population genetic patterns that accompany range expansion. We summarise a growing range of population genetic techniques that allow large changes in population size (bottlenecks and population expansions) to be inferred over a range of timescales. Finally, we illustrate some of the approaches described using data for a suite of invasions by oak gallwasps (Hymenoptera, Cynipidae, Cynipini) in Europe. We show that over timescales ranging from 500～10000 years, allele frequency data for polymorphic allozymes reveal (a) a consistent loss of genetic diversity along invasion routes, confirming the role of glacial refugia as centres of genetic diversity over these timescales, and (b) that populations in the invaded range are more subdivided genetically than those in the native range of each species. This spatial variation in population structure may be the result of variation in the patchiness of resources exploited by gallwasps, particularly host oak plants.