A systematic investigation of Ge2Fn/Ge2Fn- systems was carried out with five density functional theory (DFT) methods in conjunction with DZP++ basis sets. For each compound various structures, including minima, transition states and other energetically low lying stationary points, were optimized. The geometries and relative energies are discussed and compared. Adiabatic electron affinities, vertical electron affinities and anion vertical detachment energies are reported. Three types of dissociation energies pertaining to the global minima for each compound are reported. The theoretical predictions are in good agreement with the limited available experimental results. Many unusual structural features are predicted for these systems. Neutral Ge2F is predicted to have a bridged C2v structure, while its anion is very floppy, with the bridged structure very slightly favoured. The Ge2F2 molecule is predicted to have the butterfly structure known from experiment for Si2H2, while the Ge2F3- ion has a trans-bent structure. Ge2F3 is predicted to have an unprecedented FGe-F-GeF structure with no Ge-Ge bond, while its anion has a somewhat more conventional monobridged structure, analogous to that of the nonclassical vinyl cation. Neutral Ge2F4 has a dibridged structure of C2h symmetry, while its anion has a trans-bent structure with a very long Ge-Ge bond. The Ge2F5 molecule is doubly bridged and has no Ge-Ge bond, while the anion is of the type F2Ge-F-GeF2, again with no Ge-Ge bond. Ge2F6 has the anticipated ethane structure, as does its anion, but with a very long Ge-Ge bond. The adiabatic electron affinities (EAad) are predicted to be 2.12 (Ge2F), 2.03 (Ge2F2), 2.02 (Ge2F3), 1.64 (Ge2F4), 4.57 (Ge2F5), and 2.66 eV (Ge2F6), respectively, by the BHLYP method, which is regarded as the best method in the present paper for predicting EAs. Comparisons with the analogous C2Fn and Si2Fn systems reveals some interesting trends and differences. For example, while C2F6 will not capture an electron, Si2F6 is predicted to have small electron affinity (0.73 eV), while that of Ge2F6 is substantial. The same trend to larger EAs on going down the Periodic Table is seen for the X2F5 systems, with 1.77 (C2F5), 2.68 (Si2F5), and 4.57 eV (Ge2F5). However, the EAs do not follow a monotonic trend for the X2F, X2F3 and X2F4 systems with respect to the series X = C, Si, Ge.