The molecular structures of WF6, ReF6, OsF6, IrF6, and PtF6 have been measured by electron diffraction from the gases, the last from both PtF6 itself and from a vapor assumed to consist of a mixture of O-2 and PtF6 obtained by heating the salt O2PtF6. For models of O-h symmetry the bond lengths in the first three members of the series are essentially identical, but the Ir-F and Pt-F bonds are respectively about 0.01 and 0.02 Angstrom longer. Models of D-4h symmetry were also tested for ReF6, OsF6, and IrF6 in which operation of the Jahn-Teller effect is thought possible. For these models the same trend was seen in the average bond-length values. The effect of three-atom multiple scattering was also investigated, and experimental estimates of the effects of vibrational averaging ("shrinkage") on the distances were obtained. Normal-coordinate analyses based on the observed wavenumbers yielded stretching force constants consistent with the usual inverse bond-length/force-constant relationship. Ab initio molecular orbital optimizations of the molecules constrained to O-h symmetry were carried out at several levels of theory and basis-set size. Less extensive optimizations of ReF6, OsF6, and IrF6 with D-4h symmetry were also carried out. The best overall agreement with both the experimental values and the distance trend for O-h symmetry was obtained with the Hay-Wadt (n+1)VDZ basis on the metals and the aug-cc-pVTZ on the fluorines at the MP2 level, but these bases with B3P86 and B3PW91 density functional theory were nearly as good and with B3LYP only slightly worse. The D-4h structures for ReF6, OsF6, and IrF6 with the cited bases at the B3P86 level were slightly more stable (respectively 0.8, 2.6, and 1.4 kcal/mol) with the axial bonds shorter by about 0.04 Angstrom in ReF6 and 0.07 Angstrom in OsF6, but about 0.05 Angstrom longer in IrF6. The significance of these values is uncertain. The experimental bond lengths (r(g)/Angstrom) with estimated 2 sigma uncertainties for the models of O-h symmetry are W-F = 1.829(2), Re-F = 1.829(2), Os-F = 1.828(2), Ir-F = 1.839(2), and Pt-F = 1.852(2); the Pt-F value from the O2PtF6 sample was 1.851(2) Angstrom. Although the experimental data neither confirm nor refute the existence of the Jahn-Teller effect in ReF6, OsF6, and IrF6, they ensure that if present the distortion from O-h symmetry must be small.