Studies of bioinvasions have revealed various strategies of invasion, depending on the ecosystem invaded and the alien species concerned. Here, we consider how migration (as a demographic factor), as well as ecological and evolutionary changes, affect invasion success. We propose three main theoretical scenarios that depend on how these factors generate the match between an invader and its new environment. Our framework highlights the features that are common to, or differ among, observed invasion cases, and clarifies some general trends that have been previously highlighted in bioinvasions. We also suggest some new directions of research, such as the assessment of the time sequence of demographic, genetic and environmental changes, using detailed temporal surveys.