In this talk I will discuss our present understanding of the internal structure of Neutron Stars and the intimate relation with the equation of state of dense hadronic matter. I will examine different likely types of Neutron Stars in connection with the various possible species of hadronic stellar constituents and models for their mutual interactions. Particularly, I will consider the possibility to have deconfined quark matter in the core of Neutron Stars (Hybrid Stars), or the potential existence of compact stars entirely made of strange quark matter (Strange Stars). Finally, I will discuss some possible astrophysical consequences of the quark deconfinement phase transition in Neutron Stars.
