Myelinated axons form long-range connections that enable rapid communication between distant brain regions, but how genetics governs the strength and organization of these connections remains unclear. We perform genome-wide association studies of 206 structural connectivity measures derived from diffusion magnetic resonance imaging tractography of 26,333 UK Biobank participants, each representing the density of myelinated connections within or between a pair of cortical networks, subcortical structures or cortical hemispheres. We identify 30 independent genome-wide significant variants after Bonferroni correction for the number of measures studied (126 variants at nominal genome-wide significance) implicating genes involved in myelination (SEMA3A), neurite elongation and guidance (NUAK1, STRN, DPYSL2, EPHA3, SEMA3A, HGF, SHTN1), neural cell proliferation and differentiation (GMNC, CELF4, HGF), neuronal migration (CCDC88C), cytoskeletal organization (CTTNBP2, MAPT, DAAM1, MYO16, PLEC), and brain metal transport (SLC39A8). These variants have four broad patterns of spatial association with structural connectivity: some have disproportionately strong associations with corticothalamic connectivity, interhemispheric connectivity, or both, while others are more spatially diffuse. Structural connectivity measures are highly polygenic, with a median of 9.1 percent of common variants estimated to have non-zero effects on each measure, and exhibited signatures of negative selection. Structural connectivity measures have significant genetic correlations with a variety of neuropsychiatric and cognitive traits, indicating that connectivity-altering variants tend to influence brain health and cognitive function. Heritability is enriched in regions with increased chromatin accessibility in adult oligodendrocytes (as well as microglia, inhibitory neurons and astrocytes) and multiple fetal cell types, suggesting that genetic control of structural connectivity is partially mediated by effects on myelination and early brain development. Our results indicate pervasive, pleiotropic, and spatially structured genetic control of white-matter structural connectivity via diverse neurodevelopmental pathways, and support the relevance of this genetic control to healthy brain function.