One of the hottest topics in neuroscience is the study of brain-behavioural circuits underlying the processing of reward-related stimuli. A growing body of studies has shed new light on the neural structure of this reward system. In this paper, we discuss the significance of these studies from the perspective of a neuropsychological theory of personality, namely the Reinforcement Sensitivity Theory (RST). RST assumes that variation in sensitivity/reactivity of the reward system is the cause of individual differences in approach motivation (e.g. desire or need for achievement, persistence, and positive emotionality). Within RST, these individual differences are contained in the construct of the Behavioural Approach System (BAS). However, there is an ongoing debate as regards the nature of the BAS. This fact motivated us to review the latest refinements in the neuroscience of the BAS in the context of the reward system. In this review, we identity four distinctive aspects of the BAS: wanting, incentive motivation, striving and liking. Their behavioural effects are compared with the behavioural manifestations of testosterone, dopamine, serotonin and endogenous opioids, respectively. We conclude that the unidimensional view of the BAS is overly oversimplified; and we suggest that it should be studied as a multidimensional construct and, by implication, so too should the reward system.