Over 50 years ago, Kennedy and Higgins argued that with the lack of knowledge of the adiabatic gradient in the Earth and the slope of the melting curve of iron, the outer core could be solid and the inner core liquid. Intense experimental and theoretical activities in this field, however, have not fully solved this problem but have rather resulted in large discrepancies. I will review some of my own work related to both the temperature profile in the Earth and the melting curve of iron and will describe the experimental difficulties for measuring high-pressure/ high-temperature phase diagrams and thermal properties of the relevant materials.