  High-resolution absorption spectroscopy toward bright background sources has
had a paramount role in understanding early galaxy formation, the evolution of
the intergalactic medium and the reionisation of the Universe. However, these
studies are now approaching the boundaries of what can be achieved at
ground-based 8-10m class telescopes. The identification of primeval systems at
the highest redshifts, within the reionisation epoch and even into the dark
ages, and of the products of the first generation of stars and the chemical
enrichment of the early Universe, requires observing very faint targets with a
signal-to-noise ratio high enough to detect very faint spectral signatures. In
this paper, we describe the giant leap forward that will be enabled by ANDES,
the high-resolution spectrograph for the ELT, in these key science fields,
together with a brief, non-exhaustive overview of other extragalactic research
topics that will be pursued by this instrument, and its synergistic use with
other facilities that will become available in the early 2030s.
