The ability to form integrated circuits on flexible sheets of plastic enables attributes (for example conformal and flexible formats and lightweight and shock resistant construction) in electronic devices that are difficult or impossible to achieve with technologies that use semiconductor wafers or glass plates as substrates1. Organic small-molecule and polymer-based materials represent the most widely explored types of semiconductors for such flexible circuitry2. Although these materials and those that use films or nanostructures of inorganics have promise for certain applications, existing demonstrations of them in circuits on plastic indicate modest performance characteristics that might restrict the application possibilities. Here we report implementations of a comparatively high-performance carbon-based semiconductor consisting of sub-monolayer, random networks of single-walled carbon nanotubes to yield small- to medium-scale integrated digital circuits, composed of up to nearly 100 transistors on plastic substrates. Transistors in these integrated circuits have excellent properties: mobilities as high as 80 cm2 V-1 s-1, subthreshold slopes as low as 140 m V dec-1, operating voltages less than 5 V together with deterministic control over the threshold voltages, on/off ratios as high as 105, switching speeds in the kilohertz range even for coarse (100-m) device geometries, and good mechanical flexibility—all with levels of uniformity and reproducibility that enable high-yield fabrication of integrated circuits. Theoretical calculations, in contexts ranging from heterogeneous percolative transport through the networks to compact models for the transistors to circuit level simulations, provide quantitative and predictive understanding of these systems. Taken together, these results suggest that sub-monolayer films of single-walled carbon nanotubes are attractive materials for flexible integrated circuits, with many potential areas of application in consumer and other areas of electronics.