Experimental evidence indicates that fluid filtration through unsaturated porous media exhibits a hysteretic behavior, originating at a microscopic level from surface tension at the point of contact between water and air in the pores. As a result, the pressure-saturation constitutive relation turns out to be of hysteresis type, accurately described with the Preisach operator by a thorough fitting procedure. The main objective of the MulPHys project is to expand the knowledge about Preisach hysteresis for fluid filtration and build new mathematical models for unsaturated porous media, employing a multiscale approach. The suitability of the Preisach operator in describing the hysteretic behavior of unsaturated porous solids sparked an intense research effort in the community of experts in PDEs with hysteresis, with the goal of including the Preisach operator in mathematical models. In these endeavors, the presence of a microstructure was neglected, with the approach being directly macroscopic. Another branch of research has considered porous media as objects with a microstructure, and has derived the macroscopic description of fluid flow from local behavior. In this body of research, however, porous media are often assumed to be completely saturated so that no hysteresis can occur. With MulPHys, we will fill the gap between these two research areas, employing homogenization techniques to provide a justification of the Preisach operator as the correct tool for describing filtration. Particular attention will be paid to including gravity effects and understanding solid-liquid interactions at the microscopic level. Numerical simulations and experimental data will be instrumental in achieving these objectives. Potential applications in the preservation of historic buildings are foreseen, thus addressing a European priority and implying significant impact not only for the scientific community, but also for the professional sectors and society as a whole.