We describe the most recent data on the diffractive structure functions from the H1
and ZEUS Collaborations at HERA using four models. First, a Pomeron Structure
Function (PSF) model, in which the Pomeron is considered as an object with parton
distribution functions. Then, the Bartels Ellis Kowalski Wusthoff (BEKW) approach is
discussed, assuming the simplest perturbative description of the Pomeron using a
two-gluon ladder. A third approach, the Bialas Peschanski (BP) model, based on the
dipole formalism is then described. Finally, we discuss the Golec-Biernat-W\\\"usthoff
(GBW) saturation model which takes into account saturation effects. The best description
of all avaible measurements can be achieved with either the PSF based model or the
BEKW approach. In particular, the BEKW prediction allows to include the highest $\\beta$
measurements, which are dominated by higher twists effects and provide an efficient and
compact parametrisation of the diffractive cross section. The two other models also give
a good description of cross section measurements at small $x$ with a small number of
parameters. The comparison of all predictions allows us to identify interesting differences
in the behaviour of the effective pomeron intercept and in the shape of the longitudinal
component of the diffractive structure functions. In this last part, we present some features
that can be discriminated by new experimental measurements, completing the HERA program.
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