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We investigate the system of D-dimensional hard spheres in D-dimensional space, where D.3. For the fluid phase of these hyperspheres, we generalize scaled-particle theory to arbitrary D and furthermore use the virial expansion and the Percus-Yevick integral equation. For the crystalline phase, we adopt cell theory based on elementary geometrical assumptions about close-packed lattices. Regardless of the approximation applied, and for dimensions as high as D550, we find a first-order freezing transition, which preempts the Kirkwood second-order instability of the fluid. The relative density jump increases with D, and a generalized Lindemann rule of melting holds. We have also used ideas from fundamental-measure theory to obtain a free energy density functional for hard hyperspheres. Finally, we have calculated the surface tension of a hypersphere fluid near a hard smooth ~hyper-!wall within scaled-particle theory.