A major challenge of invasion biology is the development of a predictive framework that prevents new invasions. This is inherently difficult because different biological characteristics are important at the different stages of invasion: opportunity/transport, establishment and spread. Here, we draw from recent research on a variety of taxa to examine the evolutionary causes and consequences of biological invasions. The process of introduction may favour species with characteristics that promote success in highly disturbed, human-dominated landscapes, thus exerting novel forms of selection on introduced populations. Moreover, evidence is accumulating that multiple introductions can often be critical to the successful establishment and spread of introduced species, as they may be important sources of genetic variation necessary for adaptation in new environments or may permit the introduction of novel traits. Thus, not only should the introduction of new species be prevented, but substantial effort should also be directed to preventing the secondary introduction of previously established species (and even movement of individuals among introduced populations). Modern molecular techniques can take advantage of genetic changes postintroduction to determine the source of introduced populations and their vectors of spread, and to elucidate the mechanisms of success of some invasive species. Moreover, the growing availability of genomic tools will permit the identification of underlying genetic causes of invasive success.