[Formula Presented] and [Formula Presented] are very stable molecules with high ionization potentials and low electron affinities. Yet, we report the findings of higher aluminum hydrides, [Formula Presented] [Formula Presented] with cyclic or linear polymeric structures that are even more stable. These represent the possibilities of associating higher contents of hydrogen with aluminum clusters. We use first-principles calculations with a plane-wave basis as well as a linear combination of atomic-orbitals method. The binding energies and the highest occupied–lowest unoccupied molecular-orbital gaps of these molecules are significantly higher as compared to the values for the three-dimensional structures of hydrogenated aluminum clusters. The energetic and fragmentation behavior shows that these molecules should be stable up to a size of at least [Formula Presented] in cyclic or polymeric forms. {\textcopyright} 2003 The American Physical Society.