Cardiac fibrillation is associated with the onset of a turbulent wave activity that stops the heart from pumping. Even though this fatal arrhythmia is the leading cause of death among industrialized nations, its origin is still poorly understood and highly controversial. To a large degree, this is due to the bewildering biological complexity and architecture of the heart, which gives modelers free reign for finding multiple explanations to experimental observations that often contradict each other. Even though controversy is often a first step to progress, it seems wise to step back and to perhaps focus on simpler arrhythmogenic patterns than fibrillation, patterns for which a comparison of theory and experiment --closer to what we are accustomed to in physical systems-- seems more meaningful. This talk will discuss recent theoretical progress made in this direction. It will focus on understanding patterns of asynchronous voltage and/or calcium period-doubling oscillations (discordant alternans) observed on cellular and tissue scales. These patterns are potentially dangerous precursors to fibrillation and are in and of themselves of fundamental interest. The emergence of these patterns will be explained by (a posteriori) simple universal mechanisms that have interesting similarities, but also important differences, with know pattern formation mechanisms in other systems.