A neutron star is a compact stellar object where nuclear physics play an 
 important role. This is particularly the case of the inner crust of 
 neutron stars, composed of a lattice of nuclear clusters immersed in a 
 superfluid neutron gas. Such a system is similar to mettalic clusters present on 
 earth and requires the developpement of a band theory. In the first part 
 of my talk, I will discuss band theory and the Wigner-Seitz approximation. Then 
 I will discuss how the nuclear clusters affect the superfluid and the 
 cooling time of the inner crust. The inner crust matter is treated in the
 Wigner-Seitz approximation and the pairing properties are calculated in 
 the framework of Hartree-Fock-Bogoliubov approach.