Microwave spectra of the complexes KrAuF and KrAgBr have been measured for the first time using a cavity pulsed jet Fourier transform microwave spectrometer. The samples were prepared by laser ablation of the metal from its solid and allowing the resulting plasma to react with an appropriate precursor (Kr, plus SF6 or Br-2) contained in the backing gas of the jet (usually At). Rotational constants; geometries; centrifugal distortion constants; vibration frequencies; and Au-197, Br-79, and Br-81 nuclear quadrupole coupling constants have all been evaluated. The complexes are unusually rigid and have short Kr-Au and Kr-Ag bonds. The Au-197 nuclear quadrupole coupling constant differs radically from its value in an AuF monomer. In addition Kr-83 hyperfine structure has been measured for KrAuF and the previously reported complex KrAgF. The geometry of the latter has been reevaluated. Large values for the Kr-83 nuclear quadrupole coupling constants have been found for both complexes. Both the Au-197 and Kr-83 hyperfine constants indicate a large reorganization of the electron distribution on complex formation. A thorough assessment of the nature of the noble gas-noble metal bonding in these and related complexes (NgMX; Ng is a noble gas, M is a noble metal, and X is a halogen) has been carried out. The bond lengths are compared with sums of standard atomic and ionic radii. Ab initio calculations have produced dissociation energies along with Mulliken populations and other data on the electron distributions in the complexes. The origins of the rigidity, dissociation energies, and nuclear quadrupole coupling constants are considered. It is concluded that there is strong evidence for weak noble gas-noble metal chemical bonding in the complexes.