We have predicted a new type of noble-gas anions, XeNO2- and XeNO3-with very short Xe-N bond lengths (similar to 1.8 angstrom), using high-level electronic structure theory with extended atomic basis sets. The chemical bonding between xenon and nitrogen atoms could formally be assigned as triple bonds. The best estimates of the atomization energies of the two anions were found to be 50 and 101 kcal/mol, respectively, and the lowest unimolecular dissociation barriers were estimated to be approximately 42 kcal/mol. These anions were predicted to be kinetically stable at low temperature. The possible neutral "salts" formed between the lithium cation and these two anions were also discussed.