A new family of related borane and carborane cages has been designed computationally. These compounds obey a new electron counting rule (6m + 2n rule) rather than Wade's rule. The structures of these cages can be conceived by combining m aromatic pyramidal and n aromatic triangular units. The interstitial electrons from the m pyramids (six electrons for each unit) and the n triangles (two electrons for each unit) constitute the total 6m + 2n skeletal electrons. The greater number of skeletal electron pairs in large closo-borane cages (e.g., B32H328- or C8B24H32) achieves stabilization through the optimal occupancy of all bonding orbitals. The favorable electronic structure, the large HOMO-LUMO gaps, the large lowest positive frequencies, and the local aromaticity of the pyramidal and triangular units (as demonstrated by the large negative NICS values) of the new large closo-cages auger well for their eventual experimental realization.