The development of flexible electronic skins with high sensitivities and multimodal sensing capabilities is of great interest for applications ranging from human healthcare monitoring to robotic skins to prosthetic limbs. Although piezoresistive composite elastomers have shown great promise in this area of research, typically poor sensitivities and low response times, as well as signal drifts with temperature, have prevented further development of these materials in electronic skin applications. Here, we introduce and demonstrate a design of flexible electronic skins based on composite elastomer films that contain interlocked microdome arrays and display giant tunneling piezoresistance. Our design substantially increases the change in contact area upon loading and enables an extreme resistance-switching behavior (ROFF/RON of ∼10(5)). This translates into high sensitivity to pressure (-15.1 kPa(-1), ∼0.2 Pa minimum detection) and rapid response/relaxation times (∼0.04 s), with a minimal dependence on temperature variation. We show that our sensors can sensitively monitor human breathing flows and voice vibrations, highlighting their potential use in wearable human-health monitoring systems.