The Large Ultraviolet/Optical/Infrared Surveyor (LUVOIR) is one of four large mission concept studies led by NASA for the 2020 Decadal Survey. A versatile suite of instruments is envisioned for LUVOIR, to advance our understanding of the origin and evolution of galaxies, stars and planets that make up our Universe, and the life within it. We present POLLUX, a high-resolution spectropolarimeter, operating at UV wavelengths, designed for the 15-meter primary mirror option of LUVOIR. POLLUX study is supported by the French Space Agency (CNES) and developed by a European consortium of scientists.POLLUX will operate over a broad spectral range (98 to 390 nm), at high spectral resolution (R = 120,000). This 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 Earth-like 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. UV circular and linear polarisation 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. Since the parameter space opened by POLLUX is essentially uncharted territory, its potential for ground-breaking discoveries is high.We introduce the science case and the instrument concept, as well as the challenges offered by the development of this instrument.