The uniquely human capacity for language is supported by a distributed brain network with cerebral cortical hubs in the left inferior frontal gyrus (LIFG) and left superior temporal sulcus (LSTS). Individual differences in the graded spatial organization of function across these regions may provide sensitive phenotypes for behavioural and genetic association analyses, to better understand the biological underpinnings of language. In 41,437 UK Biobank participants we applied connectopic mapping to the LIFG and LSTS. This approach maps spatial changes across a region in terms of connectivity with the rest of the brain, based on resting state functional imaging. On average, we found that Brodmann Area (BA) 44, BA 45 and the posterior LSTS were strongly connected to the superior and inferior parietal lobe and the dorsal attention and frontoparietal networks, whereas BA 47 and the medial LSTS were strongly connected to superior frontal areas BA 8B and BA 9, and the default mode network more broadly. However, there was extensive population-wide variation in these graded connectopic patterns, which significantly mediated the effects of polygenic scores for reading ability and educational attainment on performance on a vocabulary task. Furthermore, we identified 3 genomic loci displaying multivariate association patterns with these connectopic maps, one of which coincides with a long non-coding RNA, LINC01165, with potential relevance in human evolution. Overall, connectopic mapping produced fewer significant genetic association findings and lower heritability (range from 0.3 to 9.4%) than previous parcel-based approaches for mapping language network connectivity. These observations suggest higher genetic complexity and/or prominent roles for environmental or random development