Neutral and anionic 13-atom aluminum clusters are studied with high-level, fully ab initio methods: second-order perturbation theory (MP2) and coupled cluster theory with singles, doubles, and, perturbative triples (CCSD(T)). Energies and vibrational frequencies are reported for icosahedral and decahedral isomers, and are compared with density functional theory results. At the MP2 level of theory, with all of the basis sets employed, the icosahedral structure is energetically favored over the decahedral structure for both the neutral and anionic Al-13 clusters. Hessian calculations imply that only the icosahedral structures are potential energy minima. The CCSD(T)/aug-cc-pVTZ adiabatic electron affinity of Al-13 is found to be 3.57 eV, in excellent agreement with experiment.