The cosmoIGM proposal aims at investigating the role of the Intergalactic Medium (IGM) as a cosmological probe and at exploiting the many IGM-related sinergies between observational cosmology, galaxy formation and fundamental physics. The IGM is a unique cosmological observable as it probes 3/4 of the present age of the universe, it contains up to 80% of the baryons and is sensitive to scales that are not measured by other data. In the last decade, astronomical data sets have started to be widely used by the scientific community to address important physical issues such as: the nature of the dark matter and dark energy components and their evolution; the physical properties of the baryonic matter; variation of fundamental constants; feedback processes by galaxies, etc. For example, results obtained from astronomical data are nowadays comparable to those obtained by ground based physics laboratories (e.g. neutrino masses). This proposal will rely on observations of the IGM at high and low redshift and will interpret them by means of state-of-the-art computational facilities in order to firmly establish the (yet controversial) role of the IGM as a probe for cosmology and fundamental physics. Moreover, we aim at exploring the galaxy-IGM interplay at a crucial stage of the cosmic history when the universe was few Gyrs old and star forming galaxies were strongly affecting the dynamical, thermal and chemical properties of the IGM. The hosting institution, Trieste Observatory, and the Trieste Area (ICTP, SISSA and Trieste University) have a long-standing expertise on the topics above. We foresee that the present interdisciplinary proposal will have a strong scientific impact and will help the P.I. to consolidate its independence and to create his first research team.