We demonstrate logic circuits with field-effect transistors based on single carbon nanotubes. Our device layout features local gates that provide excellent capacitive coupling between the gate and nanotube, enabling strong electrostatic doping of the nanotube from p-doping to n-doping and the study of the nonconventional long-range screening of charge along the one-dimensional nanotubes. The transistors show favorable device characteristics such as high gain (>10), a large on-off ratio (>105), and room-temperature operation. Importantly, the local-gate layout allows for integration of multiple devices on a single chip. Indeed, we demonstrate one-, two-, and three-transistor circuits that exhibit a range of digital logic operations, such as an inverter, a logic NOR, a static random-access memory cell, and an ac ring oscillator.