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As the Universe evolves, it develops a web of filamentary structure ofmatter. This cosmic web is filled with gas, with the most diffuse gas lying inthe intergalactic regions. At low redshift, the gas is predominantly warm-hot,and one of its best tracers is X-ray absorption in sightlines to backgroundquasars. In this Chapter, we present the theoretical background for theformation of the warm-hot intergalactic medium (WHIM) and present the physicalproperties of the WHIM from cosmological hydro-dynamical simulations. Wediscuss the feasibility of detecting the WHIM with X-ray absorption lines, withhigh-resolution and high signal-to-noise spectra. We present detaileddiscussion of observing techniques, including the WHIM ionization balance,observable lines, the curve of growth, and the diagnostics using the X-raylines. We present the current efforts of detecting the WHIM with gratingson-board Chandra and XMM-Newton observatories. We discuss the criticality ofWHIM detections reported in literature, where robust detections are likely fromthe circumgalactic medium of intervening galaxies, or intra-group medium,rather than truly diffuse gas in the intergalactic medium. Secure detections ofthe most diffuse gas in the low redshift large scale structure may have toawait next generation of X-ray telescopes. We end our Chapter with thediscussion of future missions carrying dispersive and non-dispersivespectrometers. We present figure-of-merit parameters for line detectibility aswell as for the number of WHIM systems that can be detected with futuremissions. These will define our ability to account for the missing low-redshiftbaryons and to understand the evolution of the Universe over half of its life.

Absorption studies of the most diffuse gas in the Large Scale Structure