The nuclear quadrupole moment of aluminum (27Al) has been re-evaluated by determining the electric field gradients at this nucleus for AlF and AlCl using CCSD(T)/aug-cc-pwCVXZ (X=T and Q) levels of theory accounting for both vibra- tional averaging and core-core/core-valence electron correlation and then comparing to the experimentally measured nuclear quadrupole coupling constants (NQCCs). The new recommended value is Q(27Al) = 148.2 ± 0.5 mb, which can be com- pared to the previous value of 146.6 ± 1 mb. Using the new value of the nuclear quadrupole moment, the accuracy is assessed for several computational approaches (i.e., HF, MP2, QCISD, CCSD, CCSD(T), and density functional theory (DFT) with the B3LYP, PBE0, and M06-2X functionals) and basis sets (the aug-cc-pVXZ and aug-cc-pwCVXZ families) for determining the nuclear quadruple coupling constants for AlCN, AlNC, AlSH, AlOH, and AlCCH, where experimental measurements are available. From the results at equilibrium geometries of the polyatomic molecules, it has been determined that (i) the CCSD(T)/aug-cc-pwCVXZ approach is needed to obtain results within 4% of the experimental measurements, (ii) typical DFT values are only within 10-15% of the experimental measurements, and (iii) the aug-cc-pVXZ family of basis sets are not recommended for computing the electric field gradients at aluminum. The present results also suggest that the NQCC for AlOH should be remeasured. Using the recommended CCSD(T)/aug-cc-pwCVXZ approach, the equilibrium geometries and corresponding NQCCs for AlCH3 and AlCCCN were determined, and the NQCCs are in excellent agreement with previously reported experimental values.