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-Wusthoff (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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