White matter tracts form the structural basis of large-scale brain networks. We applied brain-wide tractography to diffusion images from 30,810 adults (U.K. Biobank) and found significant heritability for 90 node-level and 851 edge-level network connectivity measures. Multivariate genome-wide association analyses identified 325 genetic loci, of which 80% had not been previously associated with brain metrics. Enrichment analyses implicated neurodevelopmental processes including neurogenesis, neural differentiation, neural migration, neural projection guidance, and axon development, as well as prenatal brain expression especially in stem cells, astrocytes, microglia, and neurons. The multivariate association profiles implicated 31 loci in connectivity between core regions of the left-hemisphere language network. Polygenic scores for psychiatric, neurological, and behavioral traits also showed significant multivariate associations with structural connectivity, each implicating distinct sets of brain regions with trait-relevant functional profiles. This large-scale mapping study revealed common genetic contributions to variation in the structural connectome of the human brain.