Epileptic seizures are characterised by an onset of abnormal brain activity that evolves in space and time, which ultimately returns to normal background activity. For different types of seizures, the abnormal activity can be vastly different both in duration, electrographic morphology and spatial extent. Mechanistic understanding of the different seizure dynamics (spatially, as well as temporally) is crucial for the advancement and improvement of clinical treatment. To gain a deeper mechanistic insight into different seizure dynamics, mathematical models of brain processes were developed in this thesis. These models are used to explain electrographic seizure dynamics in their temporal, as well as their spatio-temporal evolution. Our studies show that the temporal evolution of seizure dynamics can be understood in terms of prototypic waveforms, which in turn can be represented in terms of three neural population processes. Such a minimal framework lends itself to a detailed phase space analysis, which elucidates seizure waveforms and seizure transitions as topological properties of the phase space. Based on the phase space considerations we show how during spike-wave seizures, single-pulse stimuli can have more complex effects than previously thought. In terms of the spatio-temporal dynamics of seizures, mechanisms for focal seizure onset and propagation are investigated in a model cortical sheet of coupled, discretised columns. The coupling followed nearest-neighbour, as well as realistic mesoscopic cortical connectivities. Different possible causes (e.g. spatial heterogeneities) of seizure generation, as well as different seizure spreading patterns (via different networks) have been investigated. We conclude that focal seizure onset can be due to global (e.g. whole-brain level) causes, global conditions & local triggers, and local (e.g. cortical column level) causes. Clinically relevant predictions from this work include the suggestion of a specific stimulation protocol in spike-wave seizures that incorporates phase space information; and the suggestion of using microscopic cortical incisions to disrupt the integrity of abnormal cortical tissue in order to prevent focal seizure onset. In conclusion, multi-scale computational modelling of seizure dynamics is proposed as an important tool to link theoretical understanding, experimental results, and patient-specific clinical data.