Pinning models are built from discrete renewal sequences by rewarding (or
penalizing) the trajectories according to their number of renewal epochs up to
time N, and N is then sent to infinity. They are statistical mechanics
models to which a lot of attention has been paid both because they are very
relevant for applications and because of their {\sl exactly solvable
character}, while displaying a non-trivial phase transition (in fact, a
localization transition). The order of the transition depends on the tail of
the inter-arrival law of the underlying renewal and the transition is
continuous when such a tail is sufficiently heavy: this is the case on which we
will focus. The main purpose of this work is to give a mathematical treatment
of the {\sl finite size scaling limit} of pinning models, namely studying the
limit (in law) of the process close to criticality when the system size is
proportional to the correlation length
