Addiction to nicotine, alcohol and cannabis commonly co-occurs, which is thought to partly stem from a common heritable liability. To elucidate its genetic architecture, we modelled the common liability to addiction, inferred from genetic correlations among six measures of dependence and frequency of use of nicotine, alcohol and cannabis. Forty-two genetic variants were identified in the multivariate genome-wide association study on the common liability to addiction, of which 67% were novel and not associated with the six phenotypes. Mapped genes highlighted the role of dopamine (e.g., dopamine D2 gene), and showed enrichment for several components of the central nervous systems (e.g., mesocorticolimbic brain regions) and molecular pathways (dopaminergic, glutamatergic, GABAergic) that are thought to modulate drug reinforcement. Genetic correlations with other traits were most prominent for reward-related behaviours (e.g., risk-taking, cocaine and heroin use) and mood (e.g., depression, insomnia). These genome-wide results triangulate and expand previous preclinical and human studies focusing on the neurobiological substrates of addiction, and help to elucidate the common genetic architecture underlying addiction.