The thermal conductivity of the Earth's lowermost mantle controls the rate of heat flow across the core-mantle boundary, and is thus a critical parameter for determining the core and mantle thermal state and evolution. This parameter and its dependence on pressure, temperature, and composition are poorly known, in part due to the inherent difficulties in determining thermal conductivities at the high pressures and temperatures (135 GPa and 3800 K) that occur at the base of the mantle. In this dissertation I estimate the thermal conductivity of the lower mantle using measurements of the thermal conductivity of MgO and (Mg,Fe)SiO3 perovskite made at high pressure and high temperature in the laser-heated diamond anvil cell. Using three-dimensional heat flow modeling, I demonstrate that the steady-state temperature distributions that form during laser heating experiments in the diamond anvil cell depend on the sample thermal conductivity as well as the experimental geometry. Relative thermal conductivity can be determined by comparing measured temperature vs. laser power curves with a numerical model. I use this technique to determine the pressure-dependence of thermal conductivity of MgO and (Mg,Fe)SiO3 perovskite, and then I extrapolate absolute measurements of thermal conductivity taken near ambient pressure to lower mantle conditions. I also estimate the contribution of radiation to heat transfer in the lower mantle. My resulting value for the thermal conductivity of the lowermost mantle is approximately 6 W/m*K, lower than the commonly assumed value of 10 W/m*K. When combined with estimates for the lower mantle boundary layer temperature gradient, the total core-mantle boundary heat flow is roughly 7 TW. This heat flow implies a slow growth rate for the Earth's inner core.