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.