With the rapid accumulation of neuroscientific data comes a pressing need to develop models that can explain the computational processes performed by the basal ganglia. Relevant biological information spans a range of structural levels, from the activity of neuronal membranes to the role of the basal ganglia in overt behavioural control. This viewpoint presents a framework for understanding the aims, limitations and methods for testing of computational models across all structural levels. We identify distinct modelling strategies that can deliver important and complementary insights into the nature of problems the basal ganglia have evolved to solve, and describe methods that are used to solve them.