In the three lectures that I delivered at the School I tried to introduce some of the main topics in the theory of Neutron Stars where Nuclear Physics plays a major role. In the first part of these lecture notes the structure of (isolated) Neutron Stars is described briefly. The physics of the crust is first developed at basic level, with a short discussion about the properties of the Coulomb lattice of nuclei that is present both in the outer and in the inner crust. The relevance of very asymmetric nuclei is stressed, in relation with the facilities that are developing and planned throughout the world to study exotic nuclei. The many-body theory of nuclear matter is then developed, including the comparison among the different methods. Special attention is payed to neutron matter at low density, as it is present in the outer crust of Neutron Stars. Applications are developed to the theory of nuclei, by means of the energy density functional method with a microscopic basis. The many-body theory is then systematically applied to the structure of Neutron Stars. The structure of nuclei present in the lattice that forms the crust is illustrated in some detail, and a discussion is devoted to the degree of uncertainty that is present in the theory of (non-homogeneous) low density asymmetric nuclear matter. The nuclear matter EoS at higher density is discussed in relation not only to the astrophysical observations, but also to the data on heavy ion collisions at intermediate energy. The complementarity of the laboratory experiments and astrophysical observations is especially emphasized and the fruitful link between the two fields is illustrated. The problem of the maximum mass of Neutron Stars and the transition to quark matter is then introduced, and the latest observational data and their fundamental relevance are briefly discussed. It is shown that it is likely that transition to quark matter occurs at the center of massive Neutron Stars, but the overall content of quark matter is quite uncertain and model dependent. Despite these difficulties, from the latest observations it seems to emerge clearly that the quark matter EoS is more repulsive than in the simplest versions of the quark matter models usually employed to describe the high density deconfined phase. The presentation is kept at the basic level, while the more advanced developments are illustrated with specific examples. The researches that are at the frontier of the theory of Neutron Star structure are sketched, leaving to the references the detail of the theoretical developments. Unfortunately, due to the lack of time (and space) some issues, like superfluidity, are not even touched. Despite that, I hope that the present notes can stimulate the interest on these research lines and be of help to the beginners, entering this wide research field, that is promising new discoveries both at phenomenological and fundamental levels.
