The gas temperature in the cores of many clusters of galaxies drops inward by about a factor of three or more within the central 100 kpc radius. The radiative cooling time drops over the same region from 5 or more Gyr down to below a few 10~8 yr. Although it would seem that cooling flows are taking place, XMM-Newton spectra show no evidence for strong mass cooling rates of gas below 1-2 keV. Chandra images show holes coincident with radio lobes and cold fronts indicating that the core regions are complex. The observational situation is reviewed. It is likely that some form of heating is important in reducing the mass cooling rates by a factor of up to ten. Conduction of heat from the hot outer gas or heating from a central radio source are discussed, together with possible ways in which continued cooling might occur. Parallels with the formation of the baryonic part of galaxies are explored.
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