The properties of pure hadronic and hybrid compact stars are reviewed using nuclear equation of state (EoS) for β-equilibrated neutron star (NS) matter obtained using a density-dependent M3Y (DDM3Y) effective nucleonnucleon interaction. Depending on the model, the energy density of quark matter can be lower than that of this nuclear EoS at higher densities, implying the possibility of transition to quark matter inside the core and the transition density depends on the particular quark matter model used. The recent observations of the binary millisecond pulsar J1614-2230 by P.B. Demorest et al. [1] and PSR J0348+0432 by J. Antoniadis et al. [2] suggest that the masses lie within 1.97 ± 0.04 M_⊙ and 2.01 ± 0.04 M_⊙, respectively, where M_⊙ is the solar mass. In conformity with recent observations, a pure nucleonic EoS determines that the maximum mass of NS rotating with frequency ν~ 667 Hz below r-mode instability is ~ 1.95 M_⊙ with radius ~ 10 km. Compact stars with quark cores rotating with same frequency have the maximum mass of ~ 1.72 M_⊙ turns out to be lower than the observed masses.
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