The concept of aromaticity was first invented to account for the unusual stability of planar organic molecules with 4n + 2 delocalized pi electrons. Recent photoelectron spectroscopy experiments on all-metal MAI(4)(-) systems with an approximate square planar Al-4(2-) unit and an alkali metal led to the suggestion that Al-4(2-) is aromatic. The square Al-4(2-) structure was recognized as the prototype of a new family of aromatic molecules. High-level ab initio calculations based on extrapolating CCSD(T)/aug-cc-pVxZ (x = D, T, and Q) to the complete basis set limit were used to calculate the first electron affinities of Al-n, = 0-4. The calculated electron affinities, 0.41 eV (n = 0), 1.51 eV (n = 1), 1.89 eV (n = 3), and 2.18 eV (n = 4), are all in excellent agreement with available experimental data. On the basis of the high-level ab initio quantum chemical calculations, we can estimate the resonance energy and show that it is quite large, large enough to stabilize Al-4(2-) with respect to Al-4. Analysis of the calculated results shows that the aromaticity of Al-4(2-) is unusual and different from that of C6H6, Particularly, compared to the usual (1-fold) pi aromaticity in C6H6, which may be represented by two Kekule structures sharing a common a bond framework, the square Al-4(2-) structure has an unusual "multiple-fold" aromaticity determined by three independent delocalized (pi and sigma) bonding systems, each of which satisfies the 4n + 2 electron counting rule, leading to a total of 4 x 4 x 4 = 64 potential resonating Kekule-like structures without a common a frame. We also discuss the 2-fold aromaticity (pi plus sigma) of the Al-3(-) anion, which can be represented by 3 x 3 = 9 potential resonating Kekule-like structures, each with two localized chemical bonds. These results lead us to suggest a general approach (applicable to both organic and inorganic molecules) for examining delocalized chemical bonding. The possible electronic contribution to the aromaticity of a molecule should not be limited to only one particular delocalized bonding system satisfying a certain electron counting rule of aromaticity. More than one independent delocalized bonding system can simultaneously satisfy the electron counting rule of aromaticity, and therefore, a molecular structure could have multiple-fold aromaticity.