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  1. With the motivation of searching for new superconductors in the Mg–B system, we performed ab initio evolutionary searches for all the stable compounds in this binary system in the pressure range of 0–200 GPa. We found previously unknown, yet thermodynamically stable, compositions MgB 3 and Mg 3 B 10 . Experimentally known MgB 2 is stable in the entire pressure range 0–200 GPa, while MgB 7 and MgB 12 are stable at pressures below 90 GPa and 35 GPa, respectively. We predict a reentrant behavior for MgB 4 , which becomes unstable against decomposition into MgB 2 and MgB 7 at 4 GPa and then becomes stable above 61 GPa. We find ubiquity of phases with boron sandwich structures analogous to the AlB 2 -type structure. However, with the exception of MgB 2 , all other magnesium borides have low electron–phonon coupling constants λ of 0.32–0.39 and are predicted to have T c below 3 K. 
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  2. We explored the B–C–O system at pressures in the range 0–50 GPa by ab initio variable-composition evolutionary simulations in the hope of discovering new stable superhard materials. A new tetragonal thermodynamically stable phase B 4 CO 4 , space group I 4̄, and two low-enthalpy metastable compounds (B 6 C 2 O 5 , B 2 CO 2 ) have been discovered. Computed phonons and elastic constants show that these structures are dynamically and mechanically stable both at high pressure and zero pressure. B 4 CO 4 is thermodynamically stable at pressures above 23 GPa, but should remain metastable under ambient conditions. Its computed hardness is about 38–41 GPa, which suggests that B 4 CO 4 is potentially superhard. 
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