The basal ganglia are often conceptualised as three parallel domains that include all the constituent nuclei. The 'ventral domain' appears to be critical for learning flexible behaviours for exploration and foraging, as it is the recipient of converging inputs from amygdala, hippocampal formation and prefrontal cortex, putatively centres for stimulus evaluation, spatial navigation, and planning/contingency, respectively. However, compared to work on the dorsal domains, the rich potential for quantitative theories and models of the ventral domain remains largely untapped, and the purpose of this review is to provide the stimulus for this work. We systematically review the ventral domain's structures and internal organisation, and propose a functional architecture as the basis for computational models. Using a full schematic of the structure of inputs to the ventral striatum (nucleus accumbens core and shell), we argue for the existence of many identifiable processing channels on the basis of unique combinations of afferent inputs. We then identify the potential information represented in these channels by reconciling a broad range of studies from the hippocampal, amygdala and prefrontal cortex literatures with known properties of the ventral striatum from lesion, pharmacological, and electrophysiological studies. Dopamine's key role in learning is reviewed within the three current major computational frameworks; we also show that the shell-based basal ganglia sub-circuits are well placed to generate the phasic burst and dip responses of dopaminergic neurons. We detail dopamine's modulation of ventral basal ganglia's inputs by its actions on pre-synaptic terminals and post-synaptic membranes in the striatum, arguing that the complexity of these effects hint at computational roles for dopamine beyond current ideas. The ventral basal ganglia are revealed as a constellation of multiple functional systems for the learning and selection of flexible behaviours and of behavioural strategies, sharing the common operations of selection-by-disinhibition and of dopaminergic modulation.