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In this talk, I will describe adaptive numerical methods for simulating cardiac fluid mechanics, and adaptive multiscale mathematical models of electrical impulse propagation in cardiac muscle tissue. Block-structured adaptive mesh refinement (AMR) is used in both application areas to construct adaptively-refined discretizations that deploy high spatial or model resolution only where needed, e.g., in the vicinity of fluid-structure interfaces in the case of cardiac fluid mechanics, or in the vicinity of action potential wavefronts in the case of cardiac electrophysiology. Simulation results will be presented from models of blood flow in the heart, fluid-structure interaction in natural and prosthetic heart valves, and electrical impulse propagation in cardiac muscle.