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The first detections of non-virialised components in or between clusters of galaxies such as thermal emission or absorption from the WHIM or the presence of hard X-ray tails have now been made. There are several initiatives for new space missions to study the physics of the WHIM, either in emission or absorption, from the USA, Japan, Italy and The Netherlands, and new missions for studying hard X-ray tails are being designed in France, Japan and the USA. Significant theoretical progress is being made in this field, also thanks to the enormous leap in computing power for numerical models. In addition to the considerable intrinsic interest in the astrophysics of clusters, these systems are fundamental probes of the underlying cosmology and of the large-scale structure. This is the time to put the expertise of observers and theoreticians in different fields together, in order to review our current knowledge and make it available to the community in a self-contained and comprehensive - yet concise - review volume.