The cognitive decline in people with substance use disorders is well known and can be found during both the dependence and drug abstinence phases. At the clinical level, cognitive decline impairs the response to addiction treatment and increases dropout rates. It can be irreversible, even after the end of drug abuse consumption. Improving our understanding of the molecular and cellular alterations associated with cognitive decline could be essential to developing specific therapeutic strategies for its treatment. Developing animal models to simulate drug abuse-induced learning and memory alterations is critical to continue exploring this clinical situation. The main aim of this review is to summarize the most recent evidence on cognitive impairment and the associated biological markers in patients addicted to some of the most consumed drugs of abuse and in animal models simulating this clinical situation. The available information suggests the need to develop more studies to further explore the molecular alterations associated with cognitive impairment, with the ultimate goal of developing new potential therapeutic strategies.