The Earth's core is composed of iron, nickel, and a small amount of light elements (e.g., Si, S, O, C, N, H and P). The thermal conductivities of these components dominate the adiabatic heat flow in the core, which is highly correlated to geodynamo. Here we review a large number of studies on the electrical and thermal conductivity of iron and iron alloys and discuss their implications on the thermal evolution of the Earth's core. In summary, we suggest that the Wiedemann-Franz law, commonly used to convert the electrical resistivity to thermal conductivity for metals and alloys, should be cautiously applied under extremely high pressure-temperature (P-T) conditions (e.g., Earth's core) because the Lorentz number may be P-T dependent. To date, the discrepancy in the thermal conductivity of iron and iron alloys remains between those from the resistivity measurements and the thermal diffusivity modeling, where the former is systematically larger. Recent studies reconcile the electrical resistivity by first-principles calculation and direct measurements, and this is a good start in resolving this discrepancy. Due to an overall higher thermal conductivity than previously thought, the inner core age is presently constrained at similar to 1.0 Ga. However, light elements in the core would likely lower the thermal conductivity and prolong the crystallization of the inner core. Meanwhile, whether thermal convection can power the dynamo before the inner core formation depends on the amounts of the proper light elements in the core. More works are needed to establish the thermal evolution model of the core.