Evolutionary processes greatly impact the outcomes of biological invasions. An extensive body of research suggests that invasive populations often undergo phenotypic and ecological divergence from their native sources. Evolution also operates at different and distinct stages during the invasion process. Thus, it is important to incorporate evolutionary change into frameworks of biological invasions because it allows us to conceptualize how these processes may facilitate or hinder invasion success. Here, we review such processes, with an emphasis on tree invasions, and place them in the context of the unified framework for biological invasions. The processes and mechanisms described are pre-introduction evolutionary history, sampling effect, founder effect, genotype-by-environment interactions, admixture, hybridization, polyploidization, rapid evolution, epigenetics and second-genomes. For the last, we propose that co-evolved symbionts, both beneficial and harmful, which are closely physiologically associated with invasive species, contain critical genetic traits that affect the evolutionary dynamics of biological invasions. By understanding the mechanisms underlying invasion success, researchers will be better equipped to predict, understand and manage biological invasions.