Motivated by the recent discovery of rare Enormous Lyman-Alpha Nebulae (ELAN) around z~2 quasars, we have initiated a long-term observational campaign with the MUSE instrument to directly uncover the astrophysics of the gas around quasars. We present here the first 61 targets of our effort under the acronym QSO MUSEUM (Quasar Snapshot Observations with MUse: Search for Extended Ultraviolet eMission). These quasars have a median redshfit of z=3.17, absolute $i$ magnitude in the range $-29.67leq M_i(z=2)leq-27.03$, and different levels of radio-loudness. This sample unveils diverse specimens of Ly$alpha$ nebulosities extending for tens of kiloparsecs around these quasars (on average out to a maximum projected distance of 80 kpc) above a surface brightness SB$>8.8times10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ ($2sigma$). The bulk of the extended Ly$alpha$ emission is within R<50 kpc, and is characterized by relatively quiescent kinematics, with average velocity dispersions of $langle sigma_{rm Lyalpha}rangle < 400$~km~s$^{-1}$. Therefore, the motions within all these Ly$alpha$ nebulosities have amplitudes consistent with gravitational motions expected in dark matter halos hosting quasars at these redshifts, possibly reflecting the complexity in propagating a fast wind on large scales. Our current data suggest a combination of photoionization and resonant scattering as powering mechanisms of the Ly$alpha$ emission. We discover the first $zsim3$ ELAN, which confirms a very low probability ($sim1%$) of occurrence of such extreme systems at these cosmic epochs. Finally, we discuss the redshift evolution currently seen in extended Ly$alpha$ emission around radio-quiet quasars from $zsim3$ to $zsim2$, concluding that it is possibly linked to a decrease of cool gas mass within the quasars CGM from $zsim3$ to $zsim2$, and thus to the balance of cool vs hot media.
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