Means for high-density multiparametric physiological mapping and stimulation are critically important in both basic and clinical cardiology. Current conformal electronic systems are essentially 2D sheets, which cannot cover the full epicardial surface or maintain reliable contact for chronic use without sutures or adhesives. Here we create 3D elastic membranes shaped precisely to match the epicardium of the heart via the use of 3D printing, as a platform for deformable arrays of multifunctional sensors, electronic and optoelectronic components. Such integumentary devices completely envelop the heart, in a form-fitting manner, and possess inherent elasticity, providing a mechanically stable biotic/abiotic interface during normal cardiac cycles. Component examples range from actuators for electrical, thermal and optical stimulation, to sensors for pH, temperature and mechanical strain. The semiconductor materials include silicon, gallium arsenide and gallium nitride, co-integrated with metals, metal oxides and polymers, to provide these and other operational capabilities. Ex vivo physiological experiments demonstrate various functions and methodological possibilities for cardiac research and therapy.