Low-energy structures and electron affinities (EAs) for aluminum hydride clusters AlnH (n = 3-13) have been calculated using ab initio and density functional calculations. Geometries were optimized at the PBE0/def-2-TZVPP level of theory, which has been shown to match the currently accepted lowest-energy structures for the all-aluminum clusters Aln and their anions. Neutral hydride clusters with n = 4, 7, and 9-12 are predicted to adopt terminal structures with the hydrogen atom bound to only one aluminum atom and with only minor alterations of the aluminum atom arrangement from that of the all-aluminum cluster. Clusters with n = 3 and 13 are predicted to adopt "face-centered" geometries, and the n = 6 cluster is predicted to prefer an isomer with the hydrogen atom bridging two aluminum atoms, also with little or no distortion to the aluminum atom arrangement from the all-aluminum cluster. Addition of a hydrogen atom to clusters with n = 5 and 8 is predicted to distort the aluminum atom arrangement significantly from that of the corresponding all-aluminum cluster. In the anionic clusters, terminal clusters are preferred for all cluster sizes except for n = 6 that prefers a face-centered arrangement. Minor distortions in the aluminum scaffolding for Al11 and Al12 were found, while all other anionic clusters adopt structures with little or no deviation in the aluminum atom arrangement from the corresponding all-aluminum cluster. Raw adiabatic electron affinities were computed using CCSD(T)/aug-cc-pVTZ single-point energies for the anionic and neutral hydride clusters at their respective DFT geometries. Isodesmic electron affinities for the hydride clusters were computed relative to their all-aluminum counterparts and show an even-odd alternation with cluster size. Derived EAs alternate in magnitude between even- and odd-numbered clusters, with the even-numbered clusters having relatively larger EAs.