Superconducting quantum interference devices (SQUIDs) based on niobium nanobridges have been produced by means of focused ion beam milling. Typical critical currents of 4-25 A and flux sensitivities of 40-200 µV/0 were measured for sensors based on 80 nm wide, 50 nm thick, and 150 nm long bridges. A white flux noise level of 1.5 µ0/Hz1/2 was measured for a device with an area of 900 µm2 and a critical current of 15 µA. The effective area of the smallest produced SQUID was 3.6 × 10-2 µm2. Possible applications for such miniature SQUIDs are in scanning SQUID microscopy and the study of magnetic nanoparticles.