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One of the most exciting recent developments in the field of invasion biology has been the growing realisation that evolution can determine invasive species’ success. Here, we review research on contemporary evolution in invasive populations, with a focus on traits that have the potential to contribute to invasive spread. Evidence available so far indicates adaptive divergence in quantitative traits predominates, although the contribution of non-adaptive processes should not be easily discounted. Further, contemporary evolution of invasive populations appears to be more frequently spurred by abiotic factors, rather than escape from natural enemies. Important progress remains to be made on the role of hybridisation in invasion success, or the conditions under which rapid evolution of phenotypic plasticity at key traits leads to invasions. Also, we do not yet have a firm grasp on how often expansion load limits invasive spread. While convincing examples of adaptation along geographic or climatic gradients are available, we highlight conditions under which such clines would arise irrespective of biotic or abiotic conditions. We propose potentially important future lines of investigation that can illuminate the mechanistic basis of invasion success while maximising the value of invasive species for understanding evolutionary processes.