Abstract : Pairing correlations in nuclear systems play a crucial role in several aspects, i.e.
binding energies and odd-even effects, superfluid phenomena and pair transfer mechanisms, just
to quote few of them. On the theoretical side, the standard description of these features is done
by using BCS or HFB models which allow to describe in a simple way pairing effects. However,
due to the explicit breaking of the particle number, these theories present some limitations
which can be cured by using particle number projection techniques. In this paper, we will
show the merits of these techniques and their accuracy in treating pairing correlations. First, a
beyond BCS-HFB method is introduced where the effect of four quasi-particle states is included
perturbatively and subsequently the particle number is restored. We will then show the need of
restoring the good particle number also for excited states that are essential in the pair transfer
process between superfluid systems. Applications to the Richardson model are shown and
discussed.