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Clusters of galaxies are the largest gravitationally bound structures in the Universe. Their baryonic composition is dominated by hot gas that is in quasi-hydrostatic equilibrium within the dark matter dominated gravitational potential well of the cluster. The hot gas is visible through spatially extended thermal X-ray emission, and it has been studied extensively both for assessing its physical properties and also as a tracer of the large-scale structure of the Universe.

Clusters of galaxies are not isolated entities in the Universe: they are connected through a filamentary cosmic web. Theoretical predictions indicate the way this web is evolving. In the early Universe most of the gas in the web was relatively cool (~ 104 K) and visible through numerous absorption lines, designated as the so-called Lyalpha forest. In the present Universe, however, about half of all the baryons are predicted to be in a warm phase (105 - 107 K), the Warm-Hot Intergalactic Medium (WHIM), with temperatures intermediate between the hot clusters and the cool absorbing gas causing the Lyalpha forest.

The X-ray spectra of clusters are dominated by the thermal emission from the hot gas, but in some cases there appears to be evidence for hard X-ray tails or soft X-ray excesses. Hard X-ray tails are difficult to detect, and one of the topics for the team is a discussion on the significance of this detection (yet contradictory) in existing and future space experiments. Various models have been proposed to produce these hard X-ray tails, and our team reviews these processes in the context of the observational constraints in clusters.

While in some cases soft excesses in clusters can be explained as the low-energy extension of the non-thermal hard X-ray components mentioned above, there is evidence that a part may also be due to thermal emission from the WHIM. The signal seen near clusters then originates in the densest and hottest parts of the WHIM filaments, where the accelerating force of the clusters is highest and heating is strongest. A strong component of this emission is line radiation from highly ionised oxygen ions, and the role of this line emission and its observational evidence will be reviewed.

WHIM filaments not only can be observed because of their continuum or line emission, but also through absorption lines if a sufficiently strong continuum background source is present. The evidence for absorption in both UV and X-ray high-resolution spectra is discussed. Future space missions will be well adapted to study these absorption lines in more detail.

In particular in absorption lines the lower density parts between clusters become observable. In these low density regions of the WHIM not only collisional ionisation but also photo-ionisation is an important process. In general, the physics of the WHIM is challenging due to its complexity since there are many uncertain factors including the heating and cooling processes, the chemical enrichment, the role of supernova-driven bubbles or starburst winds, ram-pressure stripping, the role of shocks, magnetic fields, etc. More detailed (and sophisticated) hydrodynamical simulations with state-of-the-art spatial (and temporal) resolution are required in order to follow the impact of some (if not all) of these important processes. In particular chemical enrichment is an important process to consider as it leads to many observable predictions. We review the various physical processes relevant for the WHIM, the methods that are used to simulate this and the basic results from those models.

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