Elastic electronic surfaces will integrate stiff thin film devices onto compliant polymer substrates. These surfaces may be stretched once or many times, by up to tens of percent strain. One way to make such an elastic electronic surface is to distribute rigid subcircuit islands over the polymer surface, and then fabricate active devices on the islands. These islands need to be interconnected with stretchable metallization. We describe stretchable interconnects made of stripes of thin gold film patterned on elastomeric membranes. These membranes can be stretched by up to twice their initial length and maintain electrical conduction. We review the fabrication of these conductors, present their electrical and mechanical properties, and summarize our model for their extreme stretchability. Using such stretchable interconnects, we made the first elastic circuit, an inverter of thin film transistors. The circuit remains functional when stretched and relaxed by 12% strain.