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 modelling to genomewide 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 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.The MVP summary statistics were obtained via an approved dbGaP application (phs001672.v4.p1). The authors thank Million Veteran Program (MVP) staff, researchers, and volunteers, who have contributed to MVP, and especially participants who previously served their country in the military and now generously agreed to enroll in the study. (For details, see https://www.research.va.gov/mvp/ and Gaziano, J.M. et al. Million Veteran Program: A megabiobank to study genetic influences on health and disease. J Clin Epidemiol 70, 214-23 (2016)). This research is based on data from the Million Veteran Program, Office of Research and Development, Veterans Health Administration, and was supported by the Veterans Administration (VA) Cooperative Studies Program (CSP) award #G002. This study included summary statistics of a genetic study on cannabis use (Pasman et al, 2018 Nature Neuroscience). We would like to acknowledge all participating groups of the International Cannabis Consortium, and in particular the members of the working group including Joelle Pasman, Karin Verweij, Nathan Gillespie, Eske Derks, and Jacqueline Vink. Pasman et al, (2018) included data from the UK Biobank resource under application numbers 9905, 16406 and 25331.