We present the mission concept "Mission to Analyze the UltraViolet universE" (MAUVE), a wide-field spectrometer and imager conceived during the inaugural NASA Astrophysics Mission Design School. MAUVE responds to the 2023 Announcement of Opportunity for Probe-class missions, with a budget cap of \$1 billion, and would hypothetically launch in 2031. However, the formulation of MAUVE was an educational exercise and the mission is not being developed further. The Principal Investigator-led science of MAUVE aligns with the priorities outlined in the 2020 Astrophysics Decadal Survey, enabling new characterizations of exoplanet atmospheres, the early-time light curves of some of the universe's most explosive transients, and the poorly-understood extragalactic background light. Because the Principal Investigator science occupies 30% of the observing time available during the mission's 5 yr lifespan, we provide an observing plan that would allow for 70% of the observing time to be used for General Observer programs, with community-solicited proposals. The onboard detector (THISTLE) claims significant heritage from the Space Telescope Imaging Spectrograph on Hubble, but extends its wavelength range down to the extreme UV. We note that MAUVE would be the first satellite in decades with the ability to access this regime of the electromagnetic spectrum. MAUVE has a field of view of 900" x 900" a photometric sensitivity extending to $m_{UV}\leq 24$, and a resolving power of $R\sim1000$. This paper provides full science and mission traceability matrices for this concept, and also outlines cost and scheduling timelines aimed at enabling a within-budget mission and an on-time launch.