The Fe-based solid crystalline structure(s) present in Earth's core is(are) compact, whether hexagonal or cubic. The nature of the light element (or elements) has been the subject of considerable speculation because of its bearing on the overall bulk composition of Earth, the conditions under which the core formed, the temperature regime in the core, and the continuing processes of core-mantle reactions. While any element lighter than iron could potentially compensate the density deficit if in the proper amount, geochemical availability and solubility in the Fe-Ni alloys are the main constraining factors for the resulting real alloy. The comparisons with cosmic abundances and meteorite compositions suggest H, C, O, S, Si, and P as preferred light elements. These are authors’ main focus in this chapter. The chapter also presents the different stoichiometric compounds in the Fe-rich parts of the binary phase diagrams and their crystal structures under pressure.