A spiking neural network modeling the cerebellum is presented. The model, consisting of more than 2000 conductance-based neurons and more than 50 000 synapses, runs in real-time on a dual-processor computer. The model is implemented on an event-driven spiking neural network simulator with table-based conductance and voltage computations. The cerebellar model interacts every millisecond with a time-driven simulation of a simple environment in which adaptation experiments are setup. Learning is achieved in real-time using spike time dependent plasticity rules, which drive synaptic weight changes depending on the neurons activity and the timing in the spiking representation of an error signal. The cerebellar model is tested on learning to continuously predict a target position moving along periodical trajectories. This setup reproduces experiments with primates learning the smooth pursuit of visual targets on a screen. The model learns effectively and concurrently different target trajectories. This is true even though the spiking rate of the error representation is very low, reproducing physiological conditions. Hence, we present a complete physiologically relevant spiking cerebellar model that runs and learns in real-time in realistic conditions reproducing psychophysical experiments. This work was funded in part by the EC SpikeFORCE project (IST-2001-35271, www.spikeforce.org).