Compared to other natural sciences, the field of medicine makes little use of mathematical modeling. The sheer complexity of biological systems can explain this absence to some extent, as it made modeling prohibitively expensive in many cases. The increasing power and availability of computers, however, is making more and more biological systems accessible to numerical simulation. Today, the greatest obstacle to the application of mathematics in medicine is not the computational cost or even the mathematical difficulty of the problems, but the limited technical and mathematical background of biologists and physicians. Applied mathematicians, physicists, and engineers will have to fill this gap if medicine is to enter the twenty-first century. In this presentation I will outline the current state of the art in computer modeling of cardiac electrophysiology, and illustrate with several applied studies how scientists from different fields work together to advance our knowledge of heart disease. The heart relies on an intricate electrical activation mechanism which can be affected by diseases on many different scales, from the molecular to the tissue scale, giving rise to very interesting dynamics as well as life-threatening cardiac arrhythmias. In our research, in particular the interaction between ion-channel disease and tissue structure plays an important role. We investigate these mechanisms using multiscale models, typically running on systems with thousands of compute cores.