The structural and spectroscopic properties of the Be+–H2 and Be+–D2 electrostatic complexes are investigated theoretically. A three-dimensional ground-state potential energy surface is generated ab initio at the CCSD(T) level and used for calculating the lower rovibrational energy levels variationally. The minimum of the potential energy surface corresponds to a well depth of 3168 cm–1, an intermolecular separation of 1.776 Å, with the bond of the H2 subunit being 0.027 Å longer than for the free molecule. Taking vibrational zero point energy into account, the complexes containing para H2 and ortho D2 are predicted to have dissociation energies of 2678 and 2786 cm–1, respectively. The νHH band of Be+–H2 is predicted to be red-shifted from the free dihydrogen transition by −323 cm–1, whereas the corresponding shift for Be+–D2 is predicted to be −229 cm–1. The dissociation energy of the Be+–D2 complex is calculated to be slightly higher than the energy required to vibrationally excite the D2 subunit, raising the possibility that the onset of dissociation can be observed in the infrared predissociation spectrum at a particular rotational energy level in the νDD manifold.