Observations of redshifted Lyman α (Lyα) forest absorption in the spectra of quasistellar objects (QSOs) provide a highly sensitive probe of the distribution of gaseous matter in the universe. Over the past two decades, optical spectroscopy with large ground-based telescopes, and more recently ultraviolet (UV) spectroscopy from space, have yielded a wealth of information on what appears to be a gaseous, photoionized intergalactic medium (IGM), partly enriched by the products of stellar nucleosynthesis, residing in coherent structures over many hundreds of kiloparsecs.Recent progress with cosmological hydro-simulations based on hierarchical structure formation models has led to important insights into the physical structures giving rise to the forest. If these ideas are correct, a truly inter-and protogalactic medium [at high redshift (z ∼ 3), the main repository of baryons] collapses under the influence of dark matter gravity into flattened or filamentary structures, which are seen in absorption against background QSOs. With decreasing redshift, galaxies forming in the denser regions may contribute an increasing part of the Lyα absorption cross section. Comparisons between large data samples from the new generation of telescopes and artificial Lyα forest spectra from cosmological simulations promise to become a useful cosmological tool.