When making intertemporal decisions, i.e., decisions between outcomes occurring at different instants in time, humans and animals prefer rewards with short-term availability over rewards that become available in the long run. Discounted utility theory (DUT) is an influential normative model for intertemporal decisions that attempts to capture preference over time. It prescribes which is the best decision to take with respect to consistent, coherent and optimal choice. Over the last few decades, DUT's descriptive validity has been critically challenged. Empirical studies found systematic violations of several of DUT's assumptions, including time-consistency of preferences, stationarity, constant discounting and utility maximisation. To account for these anomalies, alternative models have been devised in various academic disciplines, including economics, psychology, biology, philosophy, and most lately, cognitive neuroscience. This article reviews the most recent literature on the behavioural and neural processes underlying intertemporal choices, and elucidates to which extent these findings can be used to explain violations of DUT's assumptions. In the first three sections, DUT is introduced, and behavioural anomalies are discussed. The fourth part focuses on the neuroscience of intertemporal choice, including its functional neuroanatomy, attempts to find a discounted value signal in the brain, and recent efforts to identify neural mechanisms producing time-inconsistencies. In the last section, the computational literature on neural learning mechanisms is reviewed. Then, a new, biologically plausible computational model of intertemporal choice is proposed that is able to explain many of the behavioural anomalies. The implications of these results help to understand why humans and animals frequently decide irrationally and to their long-term economic disadvantage, and which neural mechanisms underly such decisions.