We examined liquidus phase relations in Fe‐O ± H at ∼40 and ∼150 GPa, and in Fe‐H at 45 GPa. While it has been speculated that Fe and FeH form continuous solid solution to core pressures, our experiment on Fe‐H showed that FeH0.20 forms with the hcp structure, different from fcc for stoichiometric FeH, and melts at temperature lower than that for FeH, suggesting eutectic melting between Fe and FeH. It is consistent with the liquidus phase diagram in Fe‐O‐H, which implies the Fe‐FeH binary eutectic liquid composition of FeH0.42 at ∼40 GPa. These experiments also demonstrated the partition coefficient of H between solid Fe and liquid, DH (solid/liquid) = 0.77–0.89. We estimated the outer core liquid composition based on the liquidus phase relations, solid‐liquid partitioning, and outer/inner core densities and velocities, indicating that O and either H or Si are important core light elements. Plain Language Summary: We have investigated the melting phase relations in the Fe‐O ± H and Fe‐H systems at high pressures in a laser‐heated diamond‐anvil cell. The solid‐liquid partition coefficient of H was also determined. While it is known that Fe and stoichiometric FeH form continuous solid solution at least to ∼20 GPa, our experiments on the Fe‐O‐H ternary and Fe‐H binary systems performed at ∼40 GPa consistently suggested eutectic melting between Fe and FeH with eutectic liquid composition of FeH0.42. Based on these results, we estimated the possible range of the Earth's liquid core composition to be Fe + 2.9–5.2% O + 0.03–0.32% H + 0–3.4% Si + 1.7% S by weight, which is (a) within the liquidus field of Fe to crystallize the dense inner core, (b) compatible with seismological observations of the outer core, and (c) in chemical equilibrium with the inner core solid that explains the observed density and velocities. The results indicate that the outer core is rich in O and either H or Si, supporting the delivery of a large amount of water to the Earth found in recent planet formation theories and its sequestration into the metallic core as inferred from metal‐silicate partitioning data. Key Points: We examined the liquidus phase relations and solid/liquid partitioning in the Fe‐O ± H and Fe‐H systems at ∼40 and ∼150 GPaEutectic melting between Fe and FeH is indicated from the phase relations and melting temperature in Fe‐O‐H and Fe‐HThe outer core may include 2.9–5.2 wt% O, 0.03–0.32 wt% H, 0–3.4 wt% Si, and 1.7 wt% S [ABSTRACT FROM AUTHOR]