The thermal conductivity of iron alloys at high pressures and temperatures is a critical parameter in governing (a) the present-day heat flow out of Earth's core, (b) the inferred age of Earth's inner core, and (c) the thermal evolution of Earth's core and lowermost mantle. It is, however, one of the least well-constrained important geophysical parameters, with current estimates for end-member iron under core-mantle boundary conditions varying by about a factor of 6. Here, the current state of calculations, measurements, and inferences that constrain thermal conductivity at core conditions are reviewed. The applicability of the Wiedemann-Franz law, commonly used to convert electrical resistivity data to thermal conductivity data, is probed: Here, whether the constant of proportionality, the Lorenz number, is constant at extreme conditions is of vital importance. Electron-electron inelastic scattering and increases in Fermi-liquid-like behavior may cause uncertainties in thermal conductivities derived from both first-principles-associated calculations and electrical conductivity measurements. Additional uncertainties include the role of alloying constituents and local magnetic moments of iron in modulating the thermal conductivity. Thus, uncertainties in thermal conductivity remain pervasive, and hence a broad range of core heat flows and inner core ages appear to remain plausible.