The MESSIER Surveyor has been designed specifically to explore the extremely-low surface brightness universe at UV and optical wavelengths. The two driving science cases target the mildly and highly non-linear regimes of structure formation to test two key predictions of the LCDM scenario: (1) the detection of the putative large number of galaxy satellites, and (2) the identification of baryonic filaments in the cosmic web. The science requirements imply challenging instrumentation issues which have only recently been solved. The satellite will drift scan the entire sky in 6 bands covering the 200-1000 nm wavelength range to reach the unprecedented surface brightness levels of 34 mag/arcsec^2 in the optical and 37 mag/arcsec^2 in the UV on scales of 1 arcsec. As usual when uncovering new volumes in parameter space, many important secondary science cases will also result as free byproducts and will be discussed in some detail: the actual luminosity function of galaxies, the contribution and role of intracluster light, the fluctuations of the cosmological background radiation at UV and optical wavelengths, the warm molecular hydrogen content of galaxies at z=0.25, photometric metallicities for all GAIA stars, time-domain studies of the zodiacal light, supernovae and tidal disruption events, the chemical enrichment of the interstellar medium through mass loss of red giant stars and the accurate measure of the BAO scale at z=0.7 with over 30 million galaxies detected in Lyman-alpha at this redshift. It will provide the first space-based reference UV-optical photometric catalogue of the entire sky. Synergies with the Rubin, Roman, Euclid, SPHEREx and UVEX missions will be also discussed.