Viewed from a mathematical perspective, the heart seems simple. Waves can be initiated from an intrinsic cardiac pacemaker or an electrical stimulus delivered to the heart. Once started waves can continue propagating or can be blocked. That's it. Yet the human heart is capable of sustaining a large number of abnormal cardiac rhythms - called cardiac arrhythmias. A challenge facing basic scientists and clinicians is to derive sufficient understanding of these abnormal rhythms that an impact can be made in improving the therapy of patients with heart disease. One approach to do this is to study simple theoretical and biological models that display suprisingly complex rhythms. Some of these rhythms can be analyzed using mathematical models formulated as discrete maps (e.g. circle maps). Other rhythms can be modeled using partial differential equations for excitable media such as the FitzHugh-Nagumo equation. To study the dynamics of cardiac arrhythmias in people we use large data sets obtained from long term recordings. If we could learn to interpret this data better, it should be possible to improve the therapy for patients who are at risk of sudden cardiac death. I will summarize where we are now, and where the challenges lie.