Atrial fibrillation (AF) is the most common abnormal heart rhythm, affecting 1% of people worldwide, and is predicted to double in incidence within the next two to three decades. During AF multiple excitation wavefronts are observed to propagate continuously in atrial muscle tissue (myocardium) in an apparently disorganised manner. Ablating atrial myocardium extensively and empirically via catheter electrodes may cure AF in some cases. However, an inability to identify the specific regions critical to the persistence of AF has resulted in a failure to improve on disappointing clinical outcomes. Another clinical challenge is to understand patient variability in AF and AF progression. It is commonly thought that AF-induced tissue remodelling and age-related increases in fibrosis supports the notion that AF inexorably becomes worse. However, a recent body of clinical observations now questions how large this effect is in humans and lacks a mechanistic explanation. In this thesis, we propose that both of these clinical challenges can be addressed by better understanding the relationship between the microstructure of myocardium and the dynamics of cardiac excitation waves. We investigated this relationship using a simple mathematical model which integrates phenomenological cardiac dynamics with a structure that mimics the anisotropic, branching, cable-like structure of myocardium. As we reduced the transversal connectivity of myocardium below a threshold value, mimicking the effects of fibrosis, the creation of micro-re-entrant circuits resulted in the sudden onset of AF in the model. AF in the model was driven by periods of perpetual birth and death of these micro-re-entrant circuits. We showed that this process is determined by the presence of specific local configurations of connectivity and dysfunctional cells, so called critical regions, and the collisions between wavefronts and wavetails. The simple model reproduced many clinically observed features of AF. These included: the spontaneous onset and self-termination of AF episodes of variable duration, the observation that ablation of critical regions could sometimes terminate AF, and variability in AF behaviour and AF progression. The studies of this model presented in this thesis generate novel hypotheses regarding the mechanisms of AF and its progression, and could inform future ablation strategies.