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 (90-400 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. In this paper, we introduce the general context of LUVOIR, the design of POLLUX, and the required technology development needed to achieve the desired performances of the instrument.