Plant invasions often involve rapid evolutionary change. Founder effects, hybridization, and adaptation to novel environments cause genetic differentiation between native and introduced populations and may contribute to the success of invaders. An influential idea in this context has been the Evolution of Increased Competitive Ability (EICA) hypothesis. It proposes that after enemy release plants rapidly evolve to be less defended but more competitive, thereby increasing plant vigour in introduced populations. To detect evolutionary change in invaders, comparative studies of native versus introduced populations are needed. Here, we review the current empirical evidence from: (1) comparisons of phenotypic variation in natural populations; (2) comparisons of molecular variation with neutral genetic markers; (3) comparisons of quantitative genetic variation in a common environment; and (4) comparisons of phenotypic plasticity across different environments. Field data suggest that increased vigour and reduced herbivory are common in introduced plant populations. In molecular studies, the genetic diversity of introduced populations was not consistently different from that of native populations. Multiple introductions of invasive plants appear to be the rule rather than the exception. In tests of the EICA hypothesis in a common environment, several found increased growth or decreased resistance in introduced populations. However, few provided a full test of the EICA hypothesis by addressing growth and defence in the same species. Overall, there is reasonable empirical evidence to suggest that genetic differentiation through rapid evolutionary change is important in plant invasions. We discuss conceptual and methodological issues associated with cross-continental comparisons and make recommendations for future research. When testing for EICA, greater emphasis should be put on competitive ability and plant tolerance. Moreover, it is important to address evolutionary change in characteristics other than defence and growth that could play a role in plant invasions.