We review drug addiction from the perspective of the hypothesis that drugs of abuse interact with distinct brain memory systems. We focus on emotional and procedural forms of memory, encompassing Pavlovian and instrumental conditioning, both for action-outcome and for stimulus-response associations. Neural structures encompassed by these systems include the amygdala, hippocampus, nucleus accumbens, and dorsal striatum. Additional influences emanate from the anterior cingulate and prefrontal cortex, which are implicated in the encoding and retrieval of drug-related memories that lead to drug craving and drug use. Finally, we consider the ancillary point that chronic abuse of many drugs may impact directly on neural memory systems via neuroadaptive and neurotoxic effects that lead to cognitive impairments in which memory dysfunction is prominent.