Among the different behavioral criteria used to discriminate substance dependence (or drug addiction) from other non-disordered forms of drug use, drug intake escalation presents a number of unique features that makes it particularly suitable for modeling in nonhuman animals. This criterion has stood the passage of time despite major revisions of diagnostic systems, it is common to all known drugs of abuse and it can be readily and unambiguously operationalized in laboratory animals. Here I exhaustively review evidence showing that escalation to heavy consumption of different drugs (except perhaps nicotine) can be rapidly induced in the majority of individual animals (i.e., rats) by increased drug availability. Such an escalation of drug use is probably paralleled by an authentic escalation to drug addiction, as it is associated with the co-occurrence of other addiction-like changes (i.e., increased motivation for drug use; increased difficulty to abstain from drug use; decreased sensitivity to negative consequences). In addition, during escalation of drug intake, most individual animals become increasingly responsive to drug- and stress-primed, but apparently not cue-primed, reinstatement of drug seeking after extinction. Finally, following increased drug use, most individuals present selective cognitive dysfunctions (e.g., deficits in executive functions) that may contribute to the establishment and/or persistence of addiction. Thus, the study of individuals with escalating patterns of drug use should provide a unique and valid approach to investigate, experimentally, the behavioral and neurobiological mechanisms that underlie the progression to addiction.