Constant wavelength neutron powder diffraction data at room temperature were used to refine the crystal structures of HgBa2Can−1CunO2n+2+δ(n= 1, 2, 3, 4) superconductors, which were synthesized by solid-state reaction of the component oxides. Samples annealed both in oxygen and argon atmospheres were examined. Rietveld refinements converged to values ofRwp= 8–10% and χ2= 1.1–1.7 using a tetragonal cell withP4/mmmsymmetry. The HgBa2CuO4+δ, HgBa2CaCu2O6+δ, and HgBa2Ca2Cu3O8+δ(n= 1, 2, and 3) samples were nearly single phase, whereas the HgBa2Ca3Cu4O10+δ(n= 4) sample was primarily a mixture of then= 3 and 4 phases. For increasingn, the oxygen-annealed samples exhibitTc(diamagnetic onsets) of 94, 123, 134, and 124 K, respectively, and a contraction of the CuO2sheets (lattice parametera). The highestTcsample, HgBa2Ca2Cu3O8+δ, has the shortest apical Cu–O bond. Consistently, the atomic displacement parameter for Hg is large. For refinements in which the Hg atomic displacement parameter is fixed at a more reasonable value, occupancies of the Hg site are significantly less than full for then> 1 samples, suggesting that the Hg site is deficient and/or possibly occupied by atoms with smaller scattering lengths, e.g., Cu. The oxygen site within the Hg layer is partially occupied, and its occupation increases with increasingn. For the argon-annealed samples, the cell volume increases slightly due to the isotropic increase of both the cell parameters andTcdecreases. For HgBa2CaCu2O6+δ, the occupancy of the oxygen site within the Hg layer decreases upon annealing in argon, yet theTcincreased, implying that the initial carrier concentration is in the overdoped regime.