The wave of contraction that pumps blood from the heart is created by an electrical wave, a roughly 2-dimensional wavefront of excitation that passes through the 3-dimensional heart tissue. The most common cause of sudden cardiac death is Ventricular Fibrillation, in which the wavefront breaks up into multiple, chaotically meandering wavelets. This situation can be usefully modeled by a partial differential equation of reaction-diffusion type, which has planar wave solutions under normal conditions but also supports “turbulent” solutions under conditions that will be discussed. Nonlinear instabilities play a critical role in this transition to sudden cardiac death. Experimental results suggest that these may provide new targets for anti-fibrillatory drugs.