The basal ganglia are a set of subcortical nuclei that are believed to play a critical role in solving the problem of action selection – the competition between brain systems requesting behavioural expression through limited motor resources. Thus, the basal ganglia are hypothesised to play the role of a central “switch” that examines the salience, or urgency, of action requests and allows some actions to proceed while others are prohibited. We present a series of models of the basal ganglia (and associated circuits) that demonstrate how they might help solve the computational problem of action selection. The models range from conductance-based models of individual neurons, through spiking neuronal networks, to systems-level models with rate codes and embodied (robotic) models exhibiting behaviour. At each level of description, we seek to test the computational hypothesis that the basal ganglia implement action selection. This may be done by “mining” the known biology of these circuits for mechanisms that could perform selection. This bottom-up approach to modelling is contrasted with a complementary top-down approach that seeks to “map” abstract neuronal mechanisms for a given computation onto the biology.