The seventh and eighth virial coefficients for hard hyperspheres are calculated by Monte Carlo techniques. It is found that B-7/B-2(6)=0.001 43+/-0.000 13 and 0.000 44+/-0.000 12 in four and five dimensions, respectively, and that B-8/B-2(7)=0.000 414+/-0.000 20 in four dimensions. These values are used to investigate various proposed equations of state. Comparisons against the molecular dynamics calculations of Luban and Michels [M. Luban and J. P. J. Michels, Phys. Rev. A 41, 6796 (1990)] show that their proposed semiempirical form is excellent at higher densities. Moreover, we confirm Santos [A. Santos, J. Chem. Phys. 112, 10680 (2000)] observation in five dimensions that a suitable linear combination of the Percus-Yevick compressibility and virial equations of state fits the molecular dynamics data nearly as well as any other proposed form. (C) 2004 American Institute of Physics.