The molecular geometry of monomeric and dimeric gold trifluoride, AuF3 and Au2F6, has been determined by gas-phase electron diffraction and high-level quantum chemical calculations. Both experiment and computation indicate that the ground-state structure of AuF3 has C-2v symmetry, rather than 3-fold symmetry, with one shorter and two longer Au-F bonds and an almost T-shaped form, due to a first-order Jahn-Teller effect. CASSCF calculations show the triplet D-3h symmetry structure, (3)A', to lie about 42 kcal/mol above the (1)A(1) symmetry ground state and the D-3h symmetry singlet, (1)A', even higher than the triplet state, by about a further 13 kcal/mol. The molecule has a typical "Mexican-hat"-type potential energy surface with three equal minimum-energy structures around the brim of the hat, separated by equal-height transition structures, about 3.6 kcal/mol above the minimum energy. The geometry of the transition structure has also been calculated. The dimer has a D-2h symmetry planar, halogen-bridged geometry, with the gold atom having an approximately square-planar coordination, typical for d(8) transition metals. The geometries of AuF and Au2F2 have also been calculated. The very short Au...Au separation in Au2F2 is indicative of the so-called aurophilic interaction. This effect is much less pronounced in Au2F6.