MP2/aug-cc-pVTZ calculations were performed on triel trifluorides, ZF3 (Z=B, Al, Ga, In, Tl), and their complexes with N2 and HCN species, which are acting as Lewis bases. Interaction of the ZF3 trigonal structure with a single HCN or N2 ligand may lead to a tetrahedral structure with the tetravalent triel atom, whereas interaction of ZF3 with two ligands results in a trigonal bipyramidal structure with a pentacoordinate Z atom. Consequently, the Z atom in ZF3 compounds suffers from electron deficiency (hypovalency), it obeys the octet rule in ZF3-NCH and ZF3-N2 moieties, and it may be considered as a hypervalent center in complexes of ZF3 with two ligands. Much weaker interactions are observed for the boron complexes than for the other triel systems. This is because of the lower acidity of the B center relative to that of the other triel centers, and it may be the result of a stronger backbonding effect for BF3 than for the other triel trifluorides. Interactions in aluminum complexes are characterized by meaningful electrostatic contributions, whereas for gallium complexes, the most important electron charge shift is observed as a result of complexation. Analysis of the geometry of the triel complexes and of the Z⋅⋅⋅N interactions (triel bonds) in these complexes is based on ab initio calculations as well as on the quantum theory of atoms in molecules and the natural bond orbitals method.