A criterion is derived for selecting materials capable of withstanding the very large steady state power depositions, that occur on some areas of the vessel walls of present high temperature plasma experiments and are expected in a fusion reactor. To allow the deposited heat to be removed, the material must have a high thermal conductivity k(T). Due to the temperature gradient across the material, the removable heat flux is determined by the integral integral k(T)dT, taken between the temperature T1 on the cooling side and the maximum temperature T2 in the heat deposition area. This integral was calculated for several relevant materials. With 1 cm thick materials, the maximum steady state power flux density which can be carried off by conduction is 5 to 20 MW/m2, except for carbon materials, where values up to 90 MW/m2 can be achieved. However, the thermal conductivity of carbon materials is degraded by radiation damage.