Nuclear matter Equation of State at high density

The structure of neutron stars is determined by the nuclear Equation of State well above the nuclear saturation density. Since only limited hints from heavy ion reactions at intermediate energy can be used to constraint the Equation of State, theoretical predictions are essential. Different microscopic many-body methods for symmetric matter as well as for pure neutron matter are compared to the aim of assessing the theoretical uncertainties. The possible transition to quark matter in the interior of neutron stars is then briefly discussed and it is argued that recent observational data, if confirmed, would point towards a stiffening of the quark Equation of State with respect to the asymptotic freedom regime.