Chronic pain conditions frequently co-occur, suggesting common risks and paths to prevention and treatment. Previous studies have reported genetic correlations among specific groups of pain conditions and reported genetic risk for within-individual multisite pain counts (≤7). Here, we identified genetic risk for multiple distinct pain disorders across individuals using 24 chronic pain conditions and genomic structural equation modeling (Genomic SEM). First, we ran individual genome-wide association studies (GWASs) on all 24 conditions in the UK Biobank (N ≤ 436,000) and estimated their pairwise genetic correlations. Then we used these correlations to model their genetic factor structure in Genomic SEM, using both hypothesis- and data-driven exploratory approaches. A complementary network analysis enabled us to visualize these genetic relationships in an unstructured manner. Genomic SEM analysis revealed a general factor explaining most of the shared genetic variance across all pain conditions and a second, more specific factor explaining genetic covariance across musculoskeletal pain conditions. Network analysis revealed a large cluster of conditions and identified arthropathic, back, and neck pain as potential hubs for cross-condition chronic pain. Additionally, we ran GWASs on both factors extracted in Genomic SEM and annotated them functionally. Annotation identified pathways associated with organogenesis, metabolism, transcription, and DNA repair, with overrepresentation of strongly associated genes exclusively in brain tissues. Cross-reference with previous GWASs showed genetic overlap with cognition, mood, and brain structure. These results identify common genetic risks and suggest neurobiological and psychosocial mechanisms that should be targeted to prevent and treat cross-condition chronic pain.