The electronic and geometrical structures of the ground and low-lying excited states of the diatomic AlX and AlX- series (X = H, Li, Be, B, C, N, O, and F) are calculated by the coupled-cluster method with all singles and doubles and noniterative inclusion of triples using a large atomic natural orbital basis. All the ground-state AlX molecules except for AlF can attach an additional electron and form ground-state AlX-anions. The ground-state AlBe-, AlB-, AlC-, AlN-, and AlO-anions possess excited states that are stable toward autodetachment of an extra electron; AlBe- also has a second excited state. Low-lying excited states of all AlX but AlN can attach an extra electron and form anionic states that are stable with respect to their neutral (excited) parent states. The ground-state AlLi-, AlBe-, AlB-, AlN-, and AlO- anions are found to be thermodynamically more stable than their neutral parents. The most stable is AlO-, whose dissociation energy to Al+O- is 6.4 eV. Correspondingly, AlO possesses the largest electron affinity (2.65 eV) in the series.