An increasing emphasis has been placed on the development and use of animal models of addiction that capture defining features of human drug addiction, including escalation/binge drug use, enhanced motivation for the drug, preference for the drug over other reward options, use despite negative consequences, and enhanced drug-seeking/relapse vulnerability. The need to examine behavior in both males and females has also become apparent given evidence demonstrating that the addiction process occurs differently in males and females. This review discusses the procedures that are used to model features of addiction in animals, as well as factors that influence their development. Individual differences are also discussed, with a particular focus on sex differences. While no one procedure consistently produces all characteristics, different models have been developed to focus on certain characteristics. A history of escalating/binge patterns of use appears to be critical for producing other features characteristic of addiction, including an enhanced motivation for the drug, enhanced drug seeking, and use despite negative consequences. These characteristics tend to emerge over abstinence, and appear to increase rather than decrease in magnitude over time. In females, these characteristics develop sooner during abstinence and/or following less drug exposure as compared to males, and for psychostimulant addiction, may require estradiol. Although preference for the drug over other reward options has been demonstrated in non-human primates, it has been more difficult to establish in rats. Future research is needed to define the parameters that optimally induce each of these features of addiction in the majority of animals. Such models are essential for advancing our understanding of human drug addiction and its treatment in men and women.