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Clusters of galaxies are the largest bound systems and the most important link to the large scale structure (LSS) of the Universe. The detailed properties of clusters, such as the distributions of their various mass constituents - dark matter, hot intracluster (IC) gas, and galaxies - dynamics, and thermal structure, are of much interest both intrinsically, and for the understanding of the formation and evolution of the LSS. Moreover, detailed astrophysical knowledge of clusters is essential for their use as precise cosmological probes to measure global parameters, such as H0, OmegaM, & OmegaLambda, and parameters characterising the primordial density fluctuation field.

Recent observations of many clusters of galaxies with the Chandra and XMM satellites at energies epsilon leq 10 keV, have significantly advanced our knowledge of the morphology and thermal structure of hot IC gas, the source of the cluster thermal Bremsstrahlung emission. The improved determinations of the gas temperature, density, and metal abundances from these observations significantly improve estimates of such important quantities as the total cluster mass and its gaseous and baryonic mass fractions.

As has been the case in galaxies, in clusters too a more physically complete understanding of these systems necessitates knowledge also of non-thermal (NT) quantities and phenomena in the IC space. Observational evidence for the relevance of these phenomena in clusters comes mostly from measurements of extended regions of radio emission, and from Faraday rotation (FR) of the plane of polarisation of radio sources seen through (or inside) clusters. Since the observed radio emission is clearly synchrotron-produced, its level and spectrum yield direct information on IC relativistic electrons and magnetic fields. Information on cluster magnetic fields (separately from relativistic electron properties) is obtained also from FR measurements. Compton scattering of cosmic microwave background (CMB) photons by the radio-emitting relativistic electrons boosts photon energies to the X-and-gamma regions (e.g., Rephaeli 1979). The search for cluster NT X-ray emission has begun long ago (Rephaeli et al. 1987), but first clear indications for emission at energies epsilon geq 20 keV came only after deep dedicated observations of a few clusters with the RXTE and BeppoSAX satellites (beginning with analyses of observations of the Coma cluster (Rephaeli et al. 1999, Fusco-Femiano et al. 1999). Radio and NT X-ray observations provide quantitative measures of very appreciable magnetic fields and relativistic electron densities in the observed clusters. These results open a new dimension in the study of clusters.

This is a review of cluster NT X-ray observations and their direct implications, including prospects for the detection of gamma-ray emission. The literature on cluster NT phenomena is (perhaps somewhat surprisingly) quite extensive, including several reviews of radio emission (e.g., Govoni & Feretti 2004) and cluster magnetic fields (e.g., Carilli & Taylor 2002), and a review of the current status of radio observations by Ferrari et al. 2008 - Chapter 6, this volume. In order to properly address the comparison between magnetic field values deduced from radio observations and jointly from NT X-ray and radio measurements, we include here a brief summary of cluster radio observations. Measurements of EUV emission in a few clusters, and claims that this emission is by energetic electrons, are reviewed by Durret et al. 2008 - Chapter 4, this volume. NT radiation processes are reviewed by Petrosian et al. 2008 - Chapter 10, this volume, and relevant aspects of particle acceleration mechanisms are reviewed by Petrosian & Bykov 2008 - Chapter 11, this volume.

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