Brain structure and function have largely been studied separately in relation to the neurobiology and genetics of language. Here we used linked independent component analysis to integrate language network functional connectivity with brain volumetric and white matter structure in 32,677 UK Biobank participants, followed by analysis of behavioural, neurobiological and genetic correlates of the derived multimodal structure-function imaging components. Stronger functional connectivity between brain language areas was associated with increased volume of parts of the cerebellum and motor cortex, together with smaller ventricles and sensory parietal and occipital areas. The brain structure-function language components mediated an association between vocabulary level and polygenic scores for reading ability. We report 18 genomic loci associated with brain structure-function language components. Single-nucleotide polymorphism (SNP)-based heritability estimates for these components were 23-30%, and there was significant enrichment of heritability in primate-conserved genomic loci and fetal brain human-gained enhancer elements. This study revealed that structural correlates of functional language network connectivity extend well beyond previously defined language areas of the brain, and highlights the value of multimodal brain phenotyping for human neurogenetic discovery.