SEMINAR       Thursday, January 22th, 2004- h. 15:00  Auditorium B. Touschek     F. Burgio INFN - Catania    Equation of state of Nuclear Matter and Neutron Stars' structure           Abstract The equation of state of nuclear matter is the fundamental input to build models of neutron stars according to the Einstein's general theory of relativity. In fact, typical stellar properties like masses, radii, rotational periods, damping times of nonradial oscillations, etc., strongly depend on the equation of state at densities up to one order of magnitude larger than the saturation value. Due to the large densities attainable in the core, in massive  neutron  stars a hadron-quark phase transition is expected. In my talk, I will discuss several equations of state both for nuclear matter (Brueckner-Hartree-Fock and relativistic mean field theories), and for quark matter (Nambu--Jona-Lasinio and the MIT Bag models). I will mainly discuss the effects of hyperonic matter and quark color superconductivity on neutron stars' structure. Finally, I will propose some observational signatures, related to stellar rotations and nonradial oscillations, which could indicate the presence of deconfined quark matter in neutron stars' core.