Studies of bioinvasions have revealed various strategiesof invasion, depending on the ecosystem invaded andthe alien species concerned. Here, we considerhow migration (as a demographic factor), as well asecological and evolutionary changes, affect invasionsuccess. We propose three main theoretical scenariosthat depend on how these factors generate the matchbetween an invader and its new environment. Ourframework highlights the features that are common to,or differ among, observed invasion cases, and clarifiessome general trends that have been previously high-lighted in bioinvasions. We also suggest some newdirections of research, such as the assessment of thetime sequence of demographic, genetic and environ-mental changes, using detailed temporal surveys.