Barriers to proton transfer are computed using ab initio methods for a number of systems, charged and uncharged, that contain either intramolecular or intermolecular H-bonds. It is shown that the very strong H-bonds that occur in ionic H-bonds between electronegative atoms like O or F are characterized by proton transfer potentials that have a single symmetric minimum. The transfer barrier that separates the two minima in less strongly-bound complexes rises as the H-bond weakens, as a result of involvement of less electronegative atoms. The weakening is evident from the equilibrium H-bond lengths, as well as the calculated energetics. The strain introduced into intramolecular H-bonds can act to raise the proton transfer barrier. Nonetheless, the relationship between weaker H-bonds and lower barriers is apparent, even in excited electronic states.