The evolution of hot interstellar gas in cluster-centered cD galaxies and the inÑow of gas from the surrounding galaxy clusters are strongly coupled. Cooling Ñows arise inside the cD galaxy because the deep stellar potential and stellar mass loss increase the gas density and decrease the radiative cooling time within the galaxy. Recent X-ray observations of M87 in the Virgo Cluster and NGC 4874 in Coma reveal that the gas temperature beyond about 50 kpc from these cD galaxies is comparable to the virial temperature of the cluster, 3 or 9 keV, respectively, but within the optical galaxy the temperature drops to the galactic virial temperature D1 keV. We show that these steep thermal gradients on galactic scales follow naturally from the usual cooling inÑow assumptions without recourse to thermal conductivity. However, most of the gas must radiatively cool ("" dropout ÏÏ) before it Ñows to the galactic core ; i.e., the gas must be multiphase. The temperature and density proÐles observed in M87 and NGC 4874 can be matched with approximate gasdynamical models calculated over several gigayears with either globally uniform or centrally concentrated multiphase mass dropout. Recent XMM observations of M87 indicate single-phase Ñow at every radius with no apparent radiative cooling to low temperatures. Gasdynamical models can be made consistent with single-phase Ñow for kpc, but to avoid huge central masses r Z 10 of cooled gas, we assume that some distributed cooling dropout occurs near the center of the Ñow, where the gas temperature is T D 1 keV. The evidence in X-ray spectra for multiphase cooling beginning at lower temperatures D1 keV may be less apparent than for higher temperatures and may have escaped detection. However, even if the mass of cooled gas is distributed within kpc, it is necessary that r [ 10 the mass of cooled gas not conÑict with dynamical mass-to-light determinations. Because of small deviations from true steady-state Ñow, we Ðnd that the standard decomposition methods used by X-ray observers to determine the mass Ñow may fail rather badly, particularly when the mass dropout M 0 (r) decreases with radius in the Ñow. For this case the decomposition procedure gives the usual cooling Ñow result, which is quite unlike the true variation of in our computed models. M 0 P r, M 0 (r)