Structural preferences of single-walled and coordinatively saturated spherical and tubular nanostructures of silica have been determined by ab initio calculations. Two families of spherical (SiO2)n clusters derived from Platonic solids and Archimedean polyhedra are depicted, with n ranging from 4-120. The analogue of a truncated icosidodecahedron, Ih-symmetric Si120O240, is favored in energy, closely followed by the Ih-symmetric Si60O120-truncated icosahedron. The silica nanotubes derived from spherical clusters are capped by Si2O2 rings, whereas the tubular section consists of single oxygen bridges. Periodic studies performed with open-ended silica nanotubes and the alpha-quartz polymorph of silica, along with a comparisons to fullerenes and carbon nanotubes, suggest that tubes with diameters of approximately 1 nm should be chemically stable.