  Early X-ray observations suggested that the intracluster medium cools and
condenses at the centers of clusters, leading to a cooling flow of plasma in
the cluster core. The increased incidence of emission-line nebulosity, excess
blue light, AGN activity, and molecular gas in the cores of clusters with short
central cooling times seemed to support this idea. However, high-resolution
spectroscopic observations from XMM-Newton and Chandra have conclusively ruled
out simple, steady cooling flow models. We review the history of this subject,
the current status of X-ray observations, and some recent models that have been
proposed to explain why the core gas does not simply cool and condense.
