The introduction and spread of non-native species pose numerous threats to ecosystems worldwide.While a large body of research has focused on the ecological factors contributing to the spread of non-native species, understanding the evolutionary causes and outcomes of biological invasions is a significant knowledge gap in evolutionary ecology. A key unresolved question involves whether successful non-native species (including invasive species) are already physiologically suited or 'pre-adapted' to new environments, or if colonisation implicates novel genetic changes following introduction. Hybridisation and introgression between native and introduced species can also facilitate genetic changes, potentially displacing parental species and enhancing invasive spread. Using a comparative transcriptome approach, I investigate hypotheses regarding the role of 'preadaptation' and recent genetic change (including hybridisation) in the spread of the Mediterranean marine mussel Mytilus galloprovincialis, one of the world's most widely established introduced species. In empirical Chapter 2, I explore whether the physiological traits that make warm-tolerant and invasive M. galloprovincialis distinct from cold-tolerant and non-invasive congeners are paralleled by evolutionary divergence at the molecular level. I test the hypothesis that genomic functions experimentally associated with adaptations to temperature stress in physiological studies show accelerated evolutionary divergence in invasive M. galloprovincialis. The results of this Chapter highlight several lines of evidence that positive selection has disproportionately affected genes encoding core elements of biochemical adaptation to temperature in the M. galloprovincialis lineage, suggesting an important role for thermal adaptation in explaining interspecific genetic differences that may also mediate different propensities for invasiveness in this genus. In empirical Chapters 3 and 4, I explore post-introduction evolutionary processes associated with M. galloprovincialis introductions within Australia and around the globe. In Chapter 3, I evaluate the extent of ongoing hybridisation and introgression between introduced northern M. galloprovincialis and a morphologically indistinguishable native Australian taxon, Mytilus planulatus. Contingent on resolving the demographic history between introduced and native congeners, my results demonstrate multiple recent introductions of two divergent source lineages of M. galloprovincialis into Australia and high levels of introgression between native and introduced populations.Estimated divergence times between congeners support a separate subspecies status for native M. planulatus and shed light on intrinsic challenges for invasive species research when native and introduced species boundaries are not well-defined. In Chapter 4, I sample multiple M. galloprovincialis populations to test the hypothesis that introduced populations have experienced parallel population genetic differentiation relative to their native genomic backgro...