We summarise the science cases for an ELT that were presented in the parallel session on the intergalactic medium, and the open discussion that followed the formal presentations. Observations of the IGM with an ELT provides tremendous potential for dramatic improvements in current programmes in a very wide variety of subjects. These range from fundamental physics (expansion of the Universe, nature of the dark matter, variation of physical constants), cosmology (geometry of the Universe, large-scale structure), reionisation (ionisation state of the IGM at high $z\ge 6$), to more traditional astronomy, such as the interactions between galaxies and the IGM (metal enrichment, galactic winds and other forms of feedback), and the study of the interstellar medium in high $z$ galaxies through molecules. The requirements on ELTs and their instruments for fulfilling this potential are discussed.