The crystal and electronic structures of the entire series of alkali aluminum and alkali gallium tetrahydrides (ABH(4); A = Li, Na, K, Rb, or Cs; B = Al or Ga) are systematically investigated using an ab initio projected augmented plane-wave method. For structural stability studies, we have considered several possible structural modifications, and reproduced successfully the equilibrium structures for the known phases LiAlH4, NaAlH4, KAlH4, NaGaH4, and KGaH4. Moreover, we predict the equilibrium structures of the other unknown members of this series. RbAlH4, CsAlH4, RbGaH4, and CsGaH4 should crystallize with the KGaH4-type structure, and LiGaH4 should crystallize with the NaGaH4-type structure. According to the density of states, all these compounds have nonmetallic character with a finite band gap of around 5 eV. Charge-density plot and electron localization function analyses show that the [BH4] subunits almost look like a separate molecular species spread over the A matrix. An ionic type of interaction is present between the A and the [BH4] units. Crystal orbital Hamilton population analyses reveal that the interaction between the B and H atoms is stronger than the other interactions present in these compounds.