This paper is a theoretical study of beryllium-beryllium bonding, with emphasis on how to strengthen that bond. It deals with the structures and stabilities of several Be2Hnq Species (with n = 1, 2 and charge q = -1, 0, +1). The ground states of Be2H, Be2H+, Be2H-, Be2H2, and Be2H2- are linear, whereas that of Be2H2+ (like Be2H4) is bridged. The Be-Be bond in the linear compounds arises from sp, hybridization on Be, whereas in the bridged isomers it also has p(pi) contribution (sp2 hybridization). Hydrogenation of the weakly bound Be2 (R(e) = 4.63 bohr, omega(e) = 276 cm-1, D(e) almost-equal-to 0.1 eV) strengthens the Be-Be bond significantly, as shown in the hydrides by an average R(e)(BeBe) of 4.0 bohr for linear and of 3.8 bohr for bridged isomers. The frequencies omega-e(BeBe) lie in the range 650 +/- 50 cm-1. Hydrogenation to produce Be2H leads to a D(e)(Be-Be) of 1.24 eV, while a second hydrogenation to produce HBe-BeH strengthens it further to 3.20 eV. Both Be2H+ and Be2H- also exhibit a strong D(e)(Be-Be) of ca. 3.30 eV. The process Be2H4 (bridged) --> 2BeH2, however, is only 1.38 eV endothermic. The stability of these diberyllium hydrides mainly results from the withdrawal of antibonding charge upon Be2 effected by the attached hydrogen atoms.