The Ultraviolet Explorer (UVEX) will undertake a synoptic survey of the entire sky in near-UV (NUV) and far-UV (FUV) bands, probing the dynamic FUV and NUV universe, as well as perform a modern, all- sky imaging survey that reaches >50 times deeper than GALEX. Combined with a powerful broadband spectroscopic capability and timely response to target of opportunity discoveries, UVEX will address fundamental questions from the NASA Astrophysics Roadmap and the Astro2020 Decadal Survey, enabling unique and important studies across the breadth of astrophysics. Imaging and spectroscopic surveys with UVEX will probe key aspects of the evolution of galaxies by understanding how star formation and stellar evolution at low metallicities affect the growth and evolution of low-metallicity, low-mass galaxies in the local universe. Such galaxies contain half the mass in the local universe, and are analogs for the first galaxies, but observed at distances that make them accessible to detailed study. The UVEX time-domain surveys and prompt spectroscopic follow-up capability will probe the environments, energetics, and emission processes in the early aftermaths of gravitational wave- discovered compact object mergers, discover hot, fast UV transients, and diagnose the early stages of explosive phenomena. UVEX will become a key community resource by filling a gap in the new generation of wide-field, sensitive optical and infrared surveys provided by the Rubin, Euclid, and Roman observatories. We discuss the scientific potential of UVEX, including unique studies UVEX will enable for studying exoplanet atmospheres, hot stars, explosive phenomena, black holes, and galaxy evolution.