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Abstract: It is only very recently that neutron star (NS) observations are providing tight constraints on the dense matter equation of state (EoS): pressure, energy density, phase transition and composition, etc…. I will present a few of them, such as gravitational wave emission from NS mergers, e.g. GW170817, and x-ray emission from the surface of NS. I will also present a semi-agnostic approach for the EoS employed for such analyses. I will illustrate the complementarity between astrophysics and nuclear physics, e.g. how the confrontation to astrophysical observations (GW and X-ray) could provide additional information on the nuclear empirical parameters, such as the curvature of the symmetry energy Ksym, which might make NS another nuclear laboratory. I will also illustrate how the nuclear empirical parameters represent a way to encode our ignorance on the density dependence of the nuclear equation of state. Finally, I will present a study of phase transition towards quark matter provided by the quarkyonic model. I will conclude with preliminary results could potentially shed light on the presence of phase transition in the core of NS.