How structures of various scales formed and evolved from the early Universe up to present time is a fundamental question of astrophysical cosmology. EDGE (Piro et al., 2007) will trace the cosmic history of the baryons from the early generations of massive stars by Gamma-Ray Burst (GRB) explosions, through the period of galaxy cluster formation, down to the very low redshift Universe, when between a third and one half of the baryons are expected to reside in cosmic filaments undergoing gravitational collapse by dark matter (the so-called warm hot Spectrometer (0.1-2.2 eV) with excellent energy resolution (3 eV at 0.6 keV), a Wide-Field Imager (0.3-6 keV) with high angular resolution (HPD=15") constant over the full 1.4 degree field of view, and a Wide Field Monitor (8-200 keV) with a FOV of ¼ of the sky, which will trigger the fast repointing to the GRB. Extension of its energy response up to 1 MeV will be achieved with a GRB detector with no imaging capability. This mission is proposed to ESA as part of the Cosmic Vision call. We will outline the science drivers and describe in more detail the payload of this mission.