First-principles full potential linear muffin-tin orbital-generalized gradient approximation electronic structure calculations of the new medium-Tc superconductor (MTSC) MgB2 and related diborides indicate that superconductivity in these compounds is related to the existence of Px,y-band holes at the γ point. Based on these calculations, we explain the absence of medium-Tc superconductivity for BeB2, AlB2, ScB2, and YB2. The simulation of a number of MgB2-based ternary systems using a supercell approach demonstrates that (i) the electron doping of MgB2 (i.e., MgB2-yXy with X=Be, C, N, O) and the creation of defects in the boron sublattice (nonstoichiometric MgB2-y) are not favorable for superconductivity, and (ii) a possible way of searching for similar or higher MTSC should be via hole doping of MgB2 (CaB2) or isoelectronic substitution of Mg (i.e., Mg1-xMxB2 with M = Be, Ca, Li, Na, Cu, Zn) or creating layered superstructures of the MgB2/CaB2 type.