Using the coupled-cluster methodology and large cot-relation consistent basis sets, we have examined the BX and YBBY ((1)Sigma(g)(+)) molecules, where X = He, Ne, Ar, Kr, CO, CS, and N-2 and Y = At, Kr, CO, CS, and N-2. For the B-X series we have constructed full potential energy curves reporting total energies, equilibrium geometries, binding energies, and also spectroscopic constants for the diatomic sequence. The B-CO, B-CS, and B-N-2 ground states are of (4)Sigma(-) and (2)Pi symmetries, respectively, with the (4)Sigma(-) states having remarkably strong binding energies with respect to their adiabatic fragments. For all the triatomics the first excited P-4 state of B plays an instrumental role in the binding process, while the bonding mechanism in either the (2)Pi or (4)Sigma(-) symmetries is due to charge transfer to the empty 2p(z) orbital of the B atom. The YBBY series results by singlet coupling two B-Y (4)Sigma(-) moieties, leading to acetylene-like YBdropBY systems of (1)Sigma(g)(+) symmetry.