Invasive species are predicted to adjust their morphological, physiological and life‐history traits to adapt to their non‐native environments. Although a loss of genetic variation during invasion may restrict local adaptation, introduced species often thrive in novel environments. Despite being founded by just a few individuals, Bombus terrestris (Hymenoptera: Apidae) has in less than 30 years successfully spread across the island of Tasmania (Australia), becoming abundant and competitive with native pollinators. We use RADseq to investigate what neutral and adaptive genetic processes associated with environmental and morphological variation allow B. terrestris to thrive as an invasive species in Tasmania. Given the widespread abundance of B. terrestris, we expected little genetic structure across Tasmania and weak signatures of environmental and morphological selection. We found high gene flow with low genetic diversity, although with significant isolation‐by‐distance and spatial variation in effective migration rates. Restricted migration was evident across the mid‐central region of Tasmania, corresponding to higher elevations, pastural land, low wind speeds and low precipitation seasonality. Tajima's D indicated a recent population expansion extending from the south to the north of the island. Selection signatures were found for loci in relation to precipitation, wind speed and wing loading. Candidate loci were annotated to genes with functions related to cuticle water retention and insect flight muscle stability. Understanding how a genetically impoverished invasive bumblebee has rapidly adapted to a novel island environment provides further understanding about the evolutionary processes that determine successful insect invasions, and the potential for invasive hymenopteran pollinators to spread globally.