QSO absorption spectroscopy provides a sensitive probe of both the neutral medium and diffuse ionized gas in the distant Universe. It extends 21cm maps of gaseous structures around low-redshift galaxies both to lower gas column densities and to higher redshifts. Combining galaxy surveys with absorption-line observations of gas around galaxies enables comprehensive studies of baryon cycles in galaxy outskirts over cosmic time. This Chapter presents a review of the empirical understanding of the cosmic neutral gas reservoir from studies of damped Lya absorbers (DLAs). It describes the constraints on the star formation relation and chemical enrichment history in the outskirts of distant galaxies from DLA studies. A brief discussion of available constraints on the ionized circumgalactic gas from studies of lower column density Lya absorbers and associated ionic absorption transitions is presented at the end.
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