The electronic structure and Fermi surface of the recently discovered HgBa 2Ca n-1 Cu n O 2 n+2+δ superconductors have been determined using the full potential linear muffin-tin orbital method and precise structural information determined with neutrons by Radaelli et al. Whereas for stoichiometric HgBa 2CuO 4 (Hg-1201) the only band crossing the Fermi energy derived from the CuO antibonding state is half-filled, an additional HgO band that crosses E F exists in the case of HgBa 2CaCu 2O 6 (Hg-1212) and HgBa 2Ca 2CuO 4O 8 (Hg-1223). Thus, stoichiometric HgBa 2CuO 4 is expected to be a Mott insulator with dopants essential for forming the normal metallic state that leads to superconductivity at 95 K, in contrast to two other members of the Hg family that are expected to be “self-doped” to a metallic normal state. As in Hg-1201, the electronic structure is two-dimensional and is dominated by van Hove singularities (vHS's) to which E F is pinned by dopants whose calculated concentration is found to agree well with that determined by Radaelli et al. for Hg-1212 for their maximum T c = 128 K sample. Finally, predicted doping levels for stabilizing a large volume of the high- T c Hg-1223 phase - and hence its highest T c - are made on the basis of pinning E F to the vHS.