The development of an electronic skin is critical to the realization of artificial intelligence that comes into direct contact with humans, and to biomedical applications such as prosthetic skin. To mimic the tactile sensing properties of natural skin, large arrays of pixel pressure sensors on a flexible and stretchable substrate are required. We demonstrate flexible, capacitive pressure sensors with unprecedented sensitivity and very short response times that can be inexpensively fabricated over large areas by microstructuring of thin films of the biocompatible elastomer polydimethylsiloxane. The pressure sensitivity of the microstructured films far surpassed that exhibited by unstructured elastomeric films of similar thickness, and is tunable by using different microstructures. The microstructured films were integrated into organic field-effect transistors as the dielectric layer, forming a new type of active sensor device with similarly excellent sensitivity and response times.