The existence of a family of stable I-h-symmetric polysilane nanostructures is predicted. The smallest member of the family is the dodecahedral Si20H20 cage, followed by Sig(80)H(80), S180H180, and larger structures. The structures and stabilities of the polysilane nanostructures were determined on the basis of B3LYP and MP2 calculations. The larger cages become increasingly stable, consisting of polysilane monolayers sewed up to icosahedral shape. The structures and electronic properties of the polysilane nanostructures resemble those of the experimentally known layered polysilane, suggesting possible applications in optoelectronics.