BN alternants with barons and nitrogens fully replacing alternant sets of carbons in an alternant hydrocarbon form a special class of species of natural interest. The Huckel model for general BN alternants and certain extended Huckel models for "regular" BN alternants are shown to be simply soluble in terms of the eigensolutions for the corresponding alternant hydrocarbons. The difference in the electronegativities for boron and nitrogen implies the opening of HOMO-LUMO gaps for alternant BN clusters-or equivalently the opening of band gaps (at the Fermi energy) for BN alternant polymers. Infinite classes first of octahedral-symmetry fullerenoid cages and second of "buckytube" polymers are considered.