The brain circuitry of saccadic eye movements, from brainstem to cortex, has been extensively studied during the last 30 years. The wealth of data gathered allowed the conception of numerous computational models. These models proposed descriptions of the putative mechanisms generating this data, and, in turn, made predictions and helped to plan new experiments. In this article, we review the computational models of the five main brain regions involved in saccade generation: reticular formation saccadic burst generators, superior colliculus, cerebellum, basal ganglia and premotor cortical areas. We present the various topics these models are concerned with: location of the feedback loop, multimodal saccades, long-term adaptation, on the fly trajectory correction, strategy and metrics selection, short-term spatial memory, transformations between retinocentric and craniocentric reference frames, sequence learning, to name the principle ones. Our objective is to provide a global view of the whole system. Indeed, narrowing too much the modelled areas while trying to explain too much data is a recurrent problem that should be avoided. Moreover, beyond the multiple research topics remaining to be solved locally, questions regarding the operation of the whole structure can now be addressed by building on the existing models.