Drug addiction is a neuropsychiatric disorder with grave personal consequences that has an extraordinary global economic impact. Despite decades of research, the options available to treat addiction are often ineffective because our rudimentary understanding of drug-induced pathology in brain circuits and synaptic physiology inhibits the rational design of successful therapies. This understanding will arise first from animal models of addiction where experimentation at the level of circuits and molecular biology is possible. We will review the most common preclinical models of addictive behavior and discuss the advantages and disadvantages of each. This includes non-contingent models in which animals are passively exposed to rewarding substances, as well as widely used contingent models such as drug self-administration and relapse. For the latter, we elaborate on the different ways of mimicking craving and relapse, which include using acute stress, drug administration or exposure to cues and contexts previously paired with drug self-administration. We further describe paradigms where drug-taking is challenged by alternative rewards, such as appetitive foods or social interaction. In an attempt to better model the individual vulnerability to drug abuse that characterizes human addiction, the field has also established preclinical paradigms in which drug-induced behaviors are ranked by various criteria of drug use in the presence of negative consequences. Separation of more vulnerable animals according to these criteria, along with other innate predispositions including goal- or sign-tracking, sensation-seeking behavior or impulsivity, has established individual genetic susceptibilities to developing drug addiction and relapse vulnerability. We further examine current models of behavioral addictions such as gambling, a disorder included in the DSM-5, and exercise, mentioned in the DSM-5 but not included yet due to insufficient peer-reviewed evidence. Finally, after reviewing the face validity of the aforementioned models, we consider the most common standardized tests used by pharmaceutical companies to assess the addictive potential of a drug during clinical trials.