Biological systems are complex systems that are regulated on multiple scales, with dynamics ranging from random molecular fluctuations to spatiotemporal wave dynamics and periodic oscillations. To understand the underlying mechanisms and link the dynamics on a molecular scale to those on a tissue scale and an organ scale, the research approaches to integrating computer modeling and simulation, nonlinear dynamics, and experimental and clinical data have been widely used. In this article, we review how these approaches have been used to investigate the multiscale cardiac excitation dynamics, particularly the genesis of cardiac arrhythmias that can lead to sudden death. The specific topics covered in this review are as follows: i) mechanisms of formation of intracellular calcium sparks and waves on a subcellular scale, which can be described by the stochastic transitions between the two stable states of a bistable system and the second order phase transition, respectively; ii) mechanisms of triggered activities on a cellular scale resulting from transmembrane voltage and intracellular calcium cycling and their coupling, some of which can be well described by the bifurcation theories of the nonlinear dynamical system; iii) mechanisms for the genesis of arrhythmias on a tissue scale induced by the triggered activities, which can be regarded as dynamical instability-induced pattern formation in heterogeneous excitable media; and iv) manifestations of the excitation dynamics and transitions in the whole heart (on an organ scale) in electrocardiogram to bridge the spatiotemporal wave dynamics to clinical observations. These results indicate that nonlinear dynamics, pattern formation, and statistical physics are the fundamental components in establishing a theoretical framework for understanding cardiac arrhythmias.