In Part I observational data and hypotheses pertinent to the cooling problem of the earth are reviewed. The possible importance of heat transport by slow thermal convection currents is indicated. In Part II the theory of the cooling of a solid radioactive earth is presented by a simple method which identifies this problem with an equivalent one in the cooling of a nonradioactive earth. The facts as to slowness of cooling for temperature perturbations of long wave length are reviewed. The importance of the large heat capacity of the earth is emphasized by means of simple examples. If the earth is cooling the rate must be very small, but, if heating occurs, higher rates are possible. Thermal considerations indicate that solidification of the mantle probably occurred from the bottom up. Because the steady-state temperatures in a radioactive earth are significant in determining whether the earth is heating or cooling at depth, such temperature distributions have been computed for a dozen or more types of radioactive earths. It is probable that thermal equilibrium in the earth as a whole is far from achieved. The earth may be heating everywhere—cooling everywhere—or heating at some depths and cooling at others. The transient temperatures are expressed, and the transient heat flows computed for a number of types of suitable radioactive earths which are amenable to convenient computation. The case of heat flow in layered earths is especially examined. The observed surface heat flows fix definite upper limits to the total heat generation which may exist within about 100 or 200 kilometers of the surface. It is important to improve the reliability and scope of present information concerning (1) surface heat flows, (2) the structure and composition of the crust, and (3) the association of heat generation with crustal layers. Until better knowledge of all three of these factors in the same region is available, estimates of the amount of radioactivity existing in and below the crust will involve large uncertainties. Even granting excellent thermal information about the crust, it is possible to deduce by thermal theory only very limited information about regions below a depth of 200 or 300 km. Significant concentrations of radioactivity may exist all the way to the center without violating known criteria. At depth the problem of the cooling of the earth is nonunique to such a degree that little information about the deep interior may be deduced from purely thermal data.