Massive neutron stars (NS) are expected to possess a quark core. While the hadronic side of the NS equation of state (EOS) can be considered well established, the quark side is quite uncertain. While calculating the EOS of hadronic matter we have used the Brueckner-Bethe-Goldstone formalism with realistic two-body and three-body forces, as well as a relativistic mean field model. For quark matter we employ the MIT bag model constraining the bag constant by exploiting the recent experimental results obtained at CERN on the formation of a quark-gluon plasma. We calculate the structure of NS interiors with the EOS comprising both phases, and we find that the NS maximum masses fall in a relatively narrow interval, $1.45 M_odot leq M_{rm max} leq 1.65 M_odot$, near the lower limit of the observational range.
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