We humans interact with our physical environment through our senses of touch, sight, smell, taste, and hearing. One of the great challenges in developing artificial intelligence and natural interfaces between humans and machines is finding ways to emulate those senses through electrical circuits. Great advances have been made in creating sensors that recognize sight and sound, but the sense of touch remains difficult to engineer. In order to mimic the pressure sensing properties of natural skin, large arrays of pixel pressure sensors on a flexible and stretchable substrate are required. We developed flexible, capacitive pressure sensors with unprecedented sensitivity and very short response times that can be inexpensively fabricated over large areas by micro-structuring thin films of the dielectric elastomer polydimethylsiloxane (PDMS). The pressure sensitivity of the micro-structured films far surpasses that exhibited by unstructured elastomeric films of similar thickness and is tunable by the choice of shape and density of microstructures. The micro-structured films were integrated into organic field effect transistors as the dielectric layer, forming a new type of active sensor device with similar excellent sensitivity and response times.