POLLUX is a high-resolution, UV spectropolarimeter proposed for the 15-meter primary mirror option of LUVOIR. The instrument study is supported by the French Space Agency (CNES) and performed by a consortium of European scientists.POLLUX will operate over a broad spectral range (90 to 400 nm), at high spectral resolution (R >= 120,000), with a unique spectropolarimetric capability. It is designed to address a range of questions at the core of the LUVOIR Science portfolio.POLLUX' high resolution will permit to resolve narrow UV emission and absorption lines, thus to follow 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).The most innovative characteristic of POLLUX is its unique spectropolarimetric capability, that will enable detection of the polarized light reflected from exoplanets or from their circumplanetary material, and moons, and characterization of the magnetospheres of stars and planets, and their interactions. The magnetospheric properties of planets in the solar system will be accessible to exquisite level of details, while the influence of magnetic fields at the galactic scale and in the IGM will be measured. Circular and linear polarisation in the UV will provide a full picture of magnetic field properties, and impact for a variety of media and objects, from AGN outflows to all types of stars. It will probe the physics of accretion disks around young stars and white dwarfs, or supermassive black holes in AGNs, and constrain the properties, especially sphericity, of stellar ejecta and explosions.In this poster, we introduce the science case of POLLUX, and outline its potential for ground-breaking discoveries.