Smooth shelled mussels of the genus Mytilus are conspicuous species of many temperate intertidal environments and valued for aquaculture. However, Mytilus galloprovincialis, the native mussel of the Mediterranean, is considered one of the world’s most widespread invasive species and thrives on highly modified structures such as piers, docks, and seawalls. Extensive physiological studies show substantially greater heat tolerance in M. galloprovincialis compared its congeners, and it has been suggested that this warm adaptation contributes to its invasive success. In Australia, the best evidence for the native Mytilus planulatus (Lamarck, 1819) comes from middens that pre-date European contact, yet some genetic studies based on few loci (mtDNA, allozymes) suggest close affinity of local blue mussels to M. galloprovincialis. Examining much of the mussel gene space via RNA sequencing, we test for a molecular signal of adaptation to thermal stress in the M. galloprovincialis lineage and explore whether thermal stress related genes contribute to its propensity for invasiveness, compared to non-invasive congeners. We then investigate whether a history of invasion and introgression may explain the close genetic affinities between northern hemisphere M. galloprovincialis and M. planulatus. The complex demographic history of Australian mussels is revealed, validating the endemic taxon M. planulatus alongside hybrids derived from multiple origins of M. galloprovincialis.