Motivated by the recent discovery of rare Enormous Lyman-Alpha Nebulae ( ELAN ) around z \sim 2 quasars , we have initiated a long-term observational campaign with the MUSE/VLT 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 ( Q uasar S napshot O bservations with MU se : S earch for E xtended U ltraviolet e M ission ) . These quasars are characterized by a median redshfit of z = 3.17 ( 3.03 < z < 3.46 ) , 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 \times 10 ^ { -19 } erg s ^ { -1 } cm ^ { -2 } arcsec ^ { -2 } ( 2 \sigma ) . Irrespective of the radio-loudness of the targets , 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 _ { 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 \sim 3 ELAN , which confirms a very low probability ( \sim 1 \% ) 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 \sim 3 to z \sim 2 , concluding that it is possibly linked to a decrease of cool gas mass within the quasars ’ CGM from z \sim 3 to z \sim 2 , and thus to the balance of cool vs hot media . Overall , QSO MUSEUM opens the path to statistical and homogeneous surveys targeting the gas phases in quasars ’ halos along cosmic times .