There are many unresolved problems concerning our understanding of the Earth’s core; even fundamental properties, such as the internal structure and exact composition of the inner core, are poorly known. Although it is well established that the inner core is made of iron with some alloying elements, the structural state of the iron and the nature of the light elements involved remain uncertain. Seismic observations of the Earth’s inner core do not agree with observations from experimental and theoretical mineral physics; in particular, it seems especially tricky to simultaneously match the density, compressional- and shear-wave velocities with seismology. However, calculations on the elastic properties of Fe-Si-C ternary alloys are found to be very close to the seismic observations, suggesting that it may, finally, be possible to report a candidate core composition which is fully compatible with seismology. Another unresolved problem is the budget and distribution of water in the Earth. Current estimates have large uncertainties, most of which are due to the lack of information about the deep Earth. It has been suggested that a large amount of the water may have partitioned into the core. Calculations on the partitioning of water between iron and silicate melts indicate a siderophile nature of water at core–mantle differentiation and core–mantle boundary conditions. Although weakening with increasing temperature, water always partitions strongly into the iron liquid. Thus the Earth’s core may act as a large reservoir that contains most of the Earth’s water and hydrogen may play a role in the low density of the Earth’s core (relative to pure iron) implied by measured seismic velocities.