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Observations show that the X-ray emission from many clusters of galaxies is sharply peaked around the central brightest galaxy. The inferred radiative cooling time of the gas in that peak, where the temperature drops to the center, is much shorter than the age of the cluster, suggesting the existence of a cooling flow there (Fabian et al. 1994). X-ray spectroscopy over the past 5 yr shows that the temperature drop toward the center is limited to about a factor of three. Just when the gas should be cooling most rapidly it appears not to be cooling at all. This is sometimes known as the cooling flow problem. Careful observations show that gently distributed heat is required over a radius of up to 100 kpc to balance radiative cooling in these regions.

The issues of cooling and heating of hot gas have broad relevance to the gaseous part of galaxy formation and evolution. Brightest cluster galaxies (BCG) are the most massive galaxies known. Calculations of the clustering behaviour of cold dark matter predict a power-law mass distribution for large galaxies whereas the stellar mass observed has an exponential distribution (Benson et al. 2003). The truncation of the stellar mass distribution in massive galaxies is likely due to the process which stops cooling flows. Simple cooling flows are an ingredient of semi-analytical models for galaxy formation. The cooling of hot gas to form stars is essential for the growth of massive galaxies and cannot be studied directly for isolated systems due to Galactic absorption. The cores of galaxy clusters offer examples which can be directly observed. However they do not appear to operate in any simple manner. The problem appears to be widespread, from the most massive clusters to the centers of individual elliptical galaxies. Heating and cooling problems of hot gases are common in astronomy, with examples ranging from the interstellar medium of our own Galaxy to the Solar Corona.

The diffuse hot ionized plasma in clusters is magnetized which means that MHD processes may be important (Schekochihin et al. 2004).

Here we briefly review the main X-ray properties and emission processes of the intracluster medium (ICM) before showing the X-ray spectra of cool cores. We then discuss the main solutions which have been proposed for the cooling flow problem.