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First comprehensive review of the systems controlling delay discounting.Discounting appears to be emergent from system interactions.Discounting appears more linked to impulsivity than behavioral inhibition.The phenomenology of delay discounting (e.g. shape of the discount function; relation to mental health) has been reviewed in detail previously, but not its neural substrates. Its neuropsychology is crucial for both theory and clinical practice. So, here, we review the neural underpinnings of delay discounting. We introduce its objective summary measures; provide an atheoretical summary of current findings – linking brain regions to each objectively measurable variable; and then provide a preliminary five-stage summary model of cognitive processing; followed by a mapping of parameters to the flow of information through neural systems. The whole is designed to stimulate future research on the roles of each brain region in delay discounting. Delay and payoff produce activity in many brain areas: thalamus; sensory, parietal, temporal, cingulate, prefrontal, motor, and insular cortex; and basal ganglia. Delay discounting, then, appears to emerge from the interaction of neural systems as they process streams of events in recurrent loops and not to be a simple calculation carried out in a single center in the brain.