For photon-counting applications at ultraviolet wavelengths, there are currently no detectors that combine high efficiency (> 50%), sub-nanosecond timing resolution, and sub-Hz dark count rates. Superconducting nanowire single-photon detectors (SNSPDs) have seen success over the past decade for photon-counting applications in the near-infrared, but little work has been done to optimize SNSPDs for wavelengths below 400 nm. Here, we describe the design, fabrication, and characterization of UV SNSPDs operating at wavelengths between 250 and 370 nm. The detectors have active areas up to 56 μm in diameter, 70 - 80% efficiency at temperatures up to 4.2 K, timing resolution down to 60 ps FWHM, blindness to visible and infrared photons, and dark count rates of ∼ 0.25 counts/hr for a 56 μm diameter pixel. These performance metrics make UV SNSPDs ideal for applications in trapped-ion quantum information processing, lidar studies of the upper atmosphere, UV fluorescent-lifetime imaging microscopy, and photon-starved UV astronomy.