We design a new class of electronically stable boron nanostructures, viz.: 60n(2) boron fullerene family, boron nanotubes, and a planar boron sheet, which like the alpha-boron sheet, consists of triangular and hexagonal motifs and has symmetrically arranged hexagonal holes. The binding energy of the proposed new boron sheet is only 0.02 eV/atom lower than the alpha-boron sheet. The 60n(2) boron fullerenes are chiral and the initial icosahedral symmetry of the fullerenes is lowered due to Jahn-Teller distortion. The smallest chiral B-60 fullerene is more stable by 1.5 eV than the exact boron analogue of C-60 fullerene. (C) 2010 Elsevier B.V. All rights reserved.