In this manuscript, some relations between theoretical nuclear physics and compact stars, which are known to be excellent tools to test matter under extreme conditions, are studied. Most of these links are performed within theoretical modelling which are used to describe both nuclei and nuclear systems in astrophysics. Self-consistent approaches unifying the description of isolated nuclei, dilute nuclei in a gas of light particles, and uniform matter, are presented in this manuscript and employed to understand the physics of compact stars. The manuscript is organized as follow: The first chapter is a general introduction to impact stars and supernovae physics, as well as to nuclear physics. In the second chapter, various aspects of the modeling of the in-medium nucleon-nucleon interaction are presented, such as, extension of the Skyrme interaction in the spin and spin-isospin channels, the properties of the V(low k) interaction and the low-density properties of nuclear matter. In the third chapter, the superfluid properties of dense matter are investigated, either through an iso-vector pairing interaction design to reproduce microscopic calculations in nuclear matter, or through the impact of superfluidity on derivatives of the EOS, such as the incompressibility and the symmetry energy, and finally, is presented the use of pairing vibration to study the properties of pairing. The questions of the in-medium effective mass and its impact for the dynamics of core-collapse supernovae, and the level density in nuclei are presented in the chapter 4. The microscopic modelling of the crust of neutron stars, as derived from band theory, and including superfluidity, is shown in chapter 5. In chapter 6, some links between the properties of nuclei, and neutron stars are presented: the properties of overflowing nuclei at the neutron drip, the correlation between the curst-core properties and nuclear empirical quantities, and the stiffness of the nuclear EOS, are shown. Finally, conclusions are presented in chapter 7.