Curved surfaces, complex geometries, and time-dynamic deformations of the heart create challenges in establishing intimate, nonconstraining interfaces between cardiac structures and medical devices or surgical tools, particularly over large areas. We constructed large area designs for diagnostic and therapeutic stretchable sensor and actuator webs that conformally wrap the epicardium, establishing robust contact without sutures, mechanical fixtures, tapes, or surgical adhesives. These multifunctional web devices exploit open, mesh layouts and mount on thin, bio-resorbable sheets of silk to facilitate handling in a way that yields, after dissolution, exceptionally low mechanical moduli and thicknesses. In vivo studies in rabbit and pig animal models demonstrate the effectiveness of these device webs for measuring and spatially mapping temperature, electrophysiological signals, strain, and physical contact in sheet and balloon-based systems that also have the potential to deliver energy to perform localized tissue ablation.