Thermal conductivity controls how quickly heat can be lost from planetary interiors or crystallizing magma bodies. Thermal conductivity itself is strongly temperature-dependent, and must be measured experimentally for many different rocks and magmas, over wide temperature ranges. Our experimental results collected over 2 decades of work demonstrate some of this variability, for example that hot rocks are poor conductors of heat, as are magmas and volcanic glasses. These results imply some thermal feedbacks in processes such as mantle melting or crystallization of magma. We do not know the thermal properties of many materials and compositions relevant to the outer solar system and to exoplanets, and this needs to be addressed in the future.