Computer analysis of crystalline solar cells indicates that a substantial increase in cell conversion efficiencies can be achieved by using two-cell, multi-bandgap tandem structures instead of single-junction cells. Practical AM1 efficiencies of about 30% at one sun and over 30% at multiple suns are to be expected. The further increases in efficiency calculated for a three-cell tandem structure are smaller and may not justify the added complexity. For inexpensive two-cell tandem modules, Si is preferred for the bottom cell, and the top cell material should have a bandgap of 1.75 to 1.80 eV. The GaAs-AlAs and GaAs-GaP systems are attractive candidates for the top cell. Significant advances have been achieved in growing GaAs on Ge-coated Si substrates (for the two-terminal, two-cell structure) and in growing free-standing ultrathin GaAs layers (for the two-terminal or four-terminal structures). These material advances should be transferable to the GaAs-AlAs and GaAs-GaP systems.