The electronic and geometrical structure of AlF-, AlF2-, AlF3- and AlF4- as well as their neutral parents are determined at the unrestricted Hartree-Fock (UHF) and second-order unrestricted Moller-Plesset (UMP2) levels of theory. The results of the calculations are used for estimating the adiabatic electron affinity (E.A.) of the neutrals and fragmentation energies of both the neutrals and anions. All these characteristics were also recomputed using the UMP2/6-31+G* geometry at the fourth-order UMP4 level of theory. According to the results of the single-, double, triple-, and quadruple-excitation fourth-order Moller-Plesset (MP4SDTQ) calculations, the AlF molecule in the ground state has no positive E.A., AlF2 possesses a rather high E.A. value of 1.90 eV, and AlF3 has a smaller E.A. of 0.93 eV, whereas the last member in the series AlF4 possesses an extremely high E.A. of 7.96 eV. Such a large value is related to high stability of the anion towards fragmentation opposite to the neutral AlF4 whose dissociation energy is computed to be only 4-5 kcal/mol. It has been predicted that all the anions are rather stable towards detachment of a fluorine anion, and AlF4- is the most stable in the series. It is shown the AlF molecule is able to attach an additional electron at large interatomic distances and form a stable anion. This indicates a possibility of an increasing tendency to form resonances when the molecule is excited into a higher vibrational state. Our calculations also show that AlF in the lowest tripler state may attach an additional electron to form a metastable anion in the quartet state. This anion is predicted to be stable towards both detachment of the extra electron and dissociation. A similar increasing tendency to formation of resonance states with an additional electron with vibrational excitation is also predicted for AlF2. Also in this case, at some stretched geometries, the anion in the lowest triplet state becomes more stable than the neutral system, However, relaxation of its geometry leads to its decay to the neutral in the doublet state plus a free electron.