Substance use disorders commonly co-occur with one another and with other psychiatric disorders. They share common features including high impulsivity, negative affect, and lower executive function. We tested whether a common genetic factor undergirds liability to problematic alcohol use (PAU), problematic tobacco use (PTU), cannabis use disorder (CUD), and opioid use disorder (OUD) by applying genomic structural equation modeling to genome-wide association study summary statistics for individuals of European ancestry (Total N = 1,019,521; substance-specific Ns range: 82,707–435,563) while adjusting for the genetics of substance use (Ns = 184,765−632,802). We also tested whether shared liability across SUDs is associated with behavioral constructs (risk-taking, executive function, neuroticism; Ns = 328,339−427,037) and non-substance use psychopathology (psychotic, compulsive, and early neurodevelopmental disorders). Shared genetic liability to PAU, PTU, CUD, and OUD was characterized by a unidimensional addiction risk factor (termed The Addiction-Risk-Factor, independent of substance use. OUD and CUD demonstrated the largest loadings, while problematic tobacco use showed the lowest loading. The Addiction-Risk-Factor was associated with risk-taking, neuroticism, executive function, and non-substance psychopathology, but retained specific variance before and after accounting for the genetics of substance use. Thus, a common genetic factor partly explains susceptibility for alcohol, tobacco, cannabis, and opioid use disorder. The Addiction-Risk-Factor has a unique genetic architecture that is not shared with normative substance use or non-substance psychopathology, suggesting that addiction is not the linear combination of substance use and psychopathology.