Electronics that are capable of intimate, non-invasive integration with the soft, curvilinear surfaces of biological tissues offer important opportunities for diagnosing and treating disease and for improving brain/machine interfaces. This article describes a material strategy for a type of bio-interfaced system that relies on ultrathin electronics supported by bioresorbable substrates of silk fibroin. Mounting such devices on tissue and then allowing the silk to dissolve and resorb initiates a spontaneous, conformal wrapping process driven by capillary forces at the biotic/abiotic interface. Specialized mesh designs and ultrathin forms for the electronics ensure minimal stresses on the tissue and highly conformal coverage, even for complex curvilinear surfaces, as confirmed by experimental and theoretical studies. In vivo, neural mapping experiments on feline animal models illustrate one mode of use for this class of technology. These concepts provide new capabilities for implantable and surgical devices.