We introduce the scientific motivations for the development of the Cassegrain U-Band Efficient Spectrograph (CUBES) that is now in construction for the Very Large Telescope. The assembled cases span a broad range of contemporary topics across Solar System, Galactic and extragalactic astronomy, where observations are limited by the performance of current ground-based spectrographs shortwards of 400 nm. A brief background to each case is presented and specific technical requirements on the instrument design that flow-down from each case are identified. These were used as inputs to the CUBES design, that will provide a factor of ten gain in efficiency for astronomical spectroscopy over 300-405 nm, at resolving powers of R∼ 24,000 and ∼ 7,000. We include performance estimates that demonstrate the ability of CUBES to observe sources that are up to three magnitudes fainter than currently possible at ground-ultraviolet wavelengths, and we place its predicted performance in the context of existing facillities.
