We reviewed computer models that have been developed to reproduce and explain epileptiform activity. Unlike other already-published reviews on computer models of epilepsy, the proposed overview starts from the various types of epileptiform activity encountered during both interictal and ictal periods. Computational models proposed so far in the context of partial and generalized epilepsies are classified according to the following taxonomy: neural mass, neural field, detailed network and formal mathematical models. Insights gained about interictal epileptic spikes and high-frequency oscillations, about fast oscillations at seizure onset, about seizure initiation and propagation, about spike-wave discharges and about status epilepticus are described. This review shows the richness and complementarity of the various modeling approaches as well as the fruitful contribution of the computational neuroscience community in the field of epilepsy research. It shows that models have progressively gained acceptance and are now considered as an efficient way of integrating structural, functional and pathophysiological data about neural systems into "coherent and interpretable views". The advantages, limitations and future of modeling approaches are discussed. Perspectives in epilepsy research and clinical epileptology indicate that very promising directions are foreseen, like model-guided experiments or model-guided therapeutic strategy, among others.