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A complete enumeration is given for quasisimplicial 3-spheres with eight vertices. It is found that there are 661 spheres, and it is proved that precisely 20 of these are not polytopal.

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The enumeration of 4-poIytopes and 3-spheres with eight vertices started in [1], is completed here. We show that there are precisely 624 4-polytopes and 22 non-polytopal 3-spheres with eight vertices that are not quasisimplicial nor pyramidal. We find them all and give a detailed description of the 22 non-polytopal spheres.