Atrial fibrillation is a heart rhythm disturbance that, although characterised by a generalised derangement of electrical function causing multiple apparently “chaotic” randomly conducting electrical wavefronts throughout the atrial heart muscle (myocardium), can be cured by localised destruction (ablation) of regions critical for its maintenance. The inability to identify these specific critical targets results in the current practice of performing extremely lengthy ablation procedures of relatively indiscriminate and widespread ablation without clear end points. The clinical procedure of ablation is performed by passing thin electrodes moved around the inside of the heart, providing a measure of local electrical function inferred from the local signal (electrogram). The electrogram at any site results from summation of the local electrical function of heart muscle and its architectural topology determined by the degree of electrically inert scarring. Evolution of current ablative treatment of atrial fibrillation has been on the basis of “learning while burning” based on visual inspection of electrograms has plateaued. We need now to adopt a different perspective based on better understanding of the underlying mechanism by systematic integration of the clinically accessible information – the spatial distribution of both the scarring and the corresponding co-located electrograms – with the aim of identifying the structural and functional characteristics of regions critical for maintaining fibrillation and the targets for ablation. Although the heart has been fertile ground for mathematical modelling of the electric activity and the interaction of structure and function, mathematical modelling has had limited impact on the clinical realm of atrial fibrillation to date, but must now deliver on the promise of transformative incorporation into the targeting of ablation to improve clinical outcomes.