  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 $-29.67\leq M_i(z=2)\leq-27.03$, 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$>8.8\times10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ ($2\sigma$). 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 $\langle \sigma_{\rm Ly\alpha}\rangle < 400$~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 $z\sim3$ 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 $z\sim3$ to
$z\sim2$, concluding that it is possibly linked to a decrease of cool gas mass
within the quasars' CGM from $z\sim3$ to $z\sim2$, and thus to the balance of
cool vs hot media.
