Species invading non-native habitats can cause irreversible environmental damage and economic harm. Yet, how introduced species become widespread invaders remains poorly understood. Adaptation within native-range habitats and rapid adaptation to new environments may both influence invasion success. Here, we examine these hypotheses using 7058 SNPs from 36 native, 40 introduced and 19 farmed populations of tench, a fish native to Eurasia. We examined genetic structure among these populations and accounted for long-term evolutionary history within the native range to assess whether introduced populations exhibited lower genetic diversity than native populations. Subsequent to infer genotype-environment correlations within native-range habitats, we assessed whether adaptation to native environments may have shaped the success of some introduced populations. At the broad scale, two glacial refugia contributed to the ancestry and genomic diversity of tench. However, native, introduced and farmed populations of admixed origin exhibited up to 10-fold more genetic diversity (i.e., observed heterozygosity, expected heterozygosity and allelic richness) compared to populations with predominantly single-source ancestry. The effects of introduction to a new location were also apparent as introduced populations exhibited fewer private alleles (mean = 9.9 and 18.9 private alleles in introduced and native populations, respectively) and higher population-specific Fst compared to native populations, highlighting their distinctiveness relative to the pool of allelic frequencies across tench populations. Finally, introduced populations with varying levels of genetic variation and similar genetic compositions have become established and persisted under strikingly different climatic and ecological conditions. Our results suggest that lack of prior adaptation and low genetic variation may not consistently hinder the success of introduced populations for species with a demonstrated ability to expand their native range.