POLLUX is a high-resolution, UV spectropolarimeter proposed for the 15-meter primary mirror option of LUVOIR. The instrument Phase 0 study is supported by the French Space Agency (CNES) and performed by a consortium of European scientists. POLLUX has been designed to deliver high-resolution spectroscopy (R> 120,000) over a broad spectral range (90-390 nm). Its unique spectropolarimetric capabilities will open-up a vast new parameter space, in particular in the unexplored UV domain and in a regime where high-resolution observations with current facilities in the visible domain are severely photon starved. POLLUX will address a range of questions at the core of the LUVOIR Science portfolio. The combination of high resolution and broad coverage of the UV bandpass will resolve narrow UV emission and absorption lines originating in diffuse media, thus permitting the study of the baryon cycle over cosmic time: from galaxies forming stars out of interstellar gas and grains, and stars forming planets, to the various forms of feedback into the interstellar and intergalactic medium (ISM and IGM), and active galactic nuclei (AGN). UV circular and linear polarimetry will reveal the magnetic fields for a wide variety of objects for the first time, from AGN outflows to a diverse range of stars, stellar explosions (both supernovae and their remnants), the ISM and IGM. It will enable detection of polarized light reflected from exoplanets (or their circumplanetary material and moons), characterization of the magnetospheres of stars and planets (and their interactions), and measurements of the influence of magnetic fields at the (inter)galactic scale. In this paper, we outline the key science cases of POLLUX, together with its high-level technical requirements. The instrument design, its estimated performances, and the required technology development are presented in a separated paper.