The High-Resolution Multi-Object Spectrograph (HRMOS) is a facility instrument that we plan to propose for the Very Large Telescope (VLT) of the European Southern Observatory (ESO), following the initial presentation at the VLT 2030 workshop held at ESO in June 2019. HRMOS provides a combination of capabilities that are essential to carry out breakthrough science across a broad range of active research areas from stellar astrophysics and exoplanet studies to Galactic and Local Group archaeology. HRMOS fills a gap in capabilities amongst the landscape of future instrumentation planned for the next decade. The key characteristics of HRMOS will be high spectral resolution (R = 60000 - 80000) combined with multi-object (20-100) capabilities and long term stability that will provide excellent radial velocity precision and accuracy (10m/s). Initial designs predict that a SNR~100 will be achievable in about one hour for a star with mag(AB) = 15, while with the same exposure time a SNR~ 30 will be reached for a star with mag(AB) = 17. The combination of high resolution and multiplexing with wavelength coverage extending to relatively blue wavelengths (down to 380 nm), makes HRMOS a spectrograph that will push the boundaries of our knowledge and that is envisioned as a workhorse instrument in the future. The science cases presented in this White Paper include topics and ideas developed by the Core Science Team with the contributions from the astronomical community, also through the wide participation in the first HRMOS Workshop (https://indico.ict.inaf.it/event/1547/) that took place in Firenze (Italy) in October 2021.< Réduire