The Kappa distribution function has already gained a remarkable notoriety as one of the most suitable empirical models capable to describe suprathermal particle populations present in space plasmas, but is still controversial in achieving a statistical characterization of these nonequilibrium plasma systems based on first principles. With the proposed ISSI project our intention is to emphasize the importance of these models and to increase their visibility in the space plasma community, where the standard Maxwell(-Boltzmann) distribution function is still preferred for its simplicity, but it is not adequate for nonequilibrium collision-poor plasmas in space. Mathematically, the Kappa distribution function is a power-law generalization, and physically, it represents a realistic alternative to outline the effects of suprathermals populations by contrasting to the Maxwellian limit. Recently, we have shown that this limit must be chosen with caution, otherwise, as in the vast majority of the existing studies, the comparison may not have the expected relevance. In the observations, largely used are the simplified (1D or isotropic) models although from the rich family of Kappa distributions we can identify more specialized models to reproduce details of the observed distributions, like temperature anisotropies, (counter)streams, loss-cones and heat-fluxes. In the solar wind these are signatures of the energetic events (e.g., coronal mass ejections), which trigger critical space weather conditions. From the multitude of problematics centering on Kappa distributions, our project aims to focus on the main unresolved issues, which need urgent resolutions to be implemented in the present solar-terrestrial tasks of the ESA and NASA programs, and contribute timely to the success of the forthcoming major missions, Solar Orbiter (ESA) and Solar Probe Plus (NASA). To tackle this challenge, we propose a team of experts with a recognized experience covering three key directions in Kappa modelling, namely, observational, fundamental and specific applications. Our new collaboration is therefore expected to provide innovative analytical and numerical tools for kinetic plasma modelling of nonequibrium processes in diverse astrophysical scenarios, as well as universal concepts on their interpretation and fundamental physics to be published and accepted by the community in space sciences.