Glasses gradually dissolve and corrode when they are exposed to aqueous solutions, and for many applications it is necessary to understand and predict the kinetics of the glass dissolution. Despite the recent progress in understanding the impact of chemical composition on the dissolution rate, a detailed understanding of the structural and topological origin of chemical durability in solutions of different pH is still largely lacking. Such knowledge would enable the tailoring of glass dissolution kinetics as a function of chemical composition. In a recent study focusing on silicate minerals and glasses, a direct relation was demonstrated between the dissolution rate at high pH and the number of chemical topological constraints per atom (n_c) acting within the molecular network [Pignatelli, I.; Kumar, A.; Bauchy, M.; Sant, G. Langmuir 2016, 32, 4434–4439]. Here, we extend this work by studying the bulk dissolution rate (D_r) of a wider range of oxide glasses in various acidic, neutral, and basic solutions. The glass compositions have been selected to obtain a wide range of chemistries and values of n_c, from flexible phosphate glasses to stressed-rigid aluminosilicate glasses. We show that, in flexible glasses, the internal modes of deformation facilitate the hydration of the network, whereas, in stressed-rigid glasses, the high number of constraints largely inhibits hydration in basic, neutral, and acidic solutions. Our study of chemical dissolution also allows establishing the kinetic mechanisms, which is controlled through an effective activation energy and depends on pH and glass topology. The energy barrier that needs to be overcome to break a unit atomic constraint is approximately constant for pH > 2, but then decreases at lower pH, indicating a change in dissolution mechanism from hydrolysis to ion exchange at low pH. Thus, with this research and existing topological models, the atomistic design of new oxide glasses with a specific chemical durability for a determined pH could become possible.