Cosmochemical, geochemical, and geophysical studies indicate that Earth's core contains iron with substantial (5 to 15%) amounts of nickel. We address structural properties of pure and nickel-doped iron from both first-principles theory and experiment. In particular we show that the face-centered cubic (fcc) phase may become as stable as the hexagonal close-packed (hcp) phase at pressures around 300-360 GPa and temperatures around 5000-6000 K, corresponding to the Earth's core conditions. Our temperature-quenched experiments indicate that the face-centered cubic (fcc) phase of pure iron can exist in the pressure-temperature region above 160 GPa and 3700 K. Further, we show that at pressures above 225 GPa and temperatures over 3400 K, Fe0.9Ni0.1 alloy adopts the body-centered cubic (bcc) structure