  We review the X-ray spectra of the cores of clusters of galaxies. Recent high
resolution X-ray spectroscopic observations have demonstrated a severe deficit
of emission at the lowest X-ray temperatures as compared to that expected from
simple radiative cooling models. The same observations have provided compelling
evidence that the gas in the cores is cooling below half the maximum
temperature. We review these results, discuss physical models of cooling
clusters, and describe the X-ray instrumentation and analysis techniques used
to make these observations. We discuss several viable mechanisms designed to
cancel or distort the expected process of X-ray cluster cooling.
