Arcus is a high-resolution soft X-ray and far-ultraviolet spectroscopy mission submitted to the National Aeronautics and Space Administration’s inaugural Astrophysics Probe solicitation. Arcus makes simultaneous observations in these two critical wavelength regimes to address a broad range of science questions highlighted by the 2020 Astronomy and Astrophysics Decadal Survey, from the temperature and composition of the missing baryons in the intergalactic medium to the evolution of stars and their influence on orbiting planets. We present the science motivation for and performance of the Arcus ultraviolet spectrograph (UVS). UVS comprises a 60-cm, off-axis Cassegrain telescope feeding an imaging spectrograph operating over the 970- to 1580-Å bandpass. The instrument employs two interchangeable diffraction gratings to provide medium-resolution spectroscopy (R>20,000 in two grating modes centered at ∼1110 and 1390 Å). The spectra are recorded on an open-face, photon-counting microchannel plate detector. The instrument design achieves an end-to-end sensitivity >10 times that of the Far-Ultraviolet Spectroscopic Explorer over the key 1020- to 1150-Å range and offers arcsecond-level angular resolution spectral imaging over a 6-arcminute-long slit for observations of extended sources. We describe the example science investigations for far-ultraviolet spectroscopy on Arcus, the resultant instrument design and predicted performance, and simulated data from potential General Observer programs with Arcus.