Pairing interaction is fundamental in modern nuclear physics. We explore it in terms of the Bardeen-Cooper-Schrieffer (BCS) theory, adapted to nuclear systems. We discuss both the benefits and drawbacks, by deriving the relevant equations in the framework of minimisation, equation of motion, elimination of dangerous graph and time dependent procedures. Moreover, we try to address the errors coming from the non-conservation of particles in the system by comparing the approximate Lipkin-Nogami method to the exact Richardson solution. We describe alpha clustering in terms of the "isospin pairing" between proton and neutron pairs. Finally, we perform a system analysis of the pairing Hamiltonian in terms of dispersion relations for pairing vibrations.