The role of H bonds in the formation of the fragility and dielectric properties of highly viscous liquids is investigated. The heuristic supposition about the proportionality between the logarithm of the shear viscosity and oscillatory contributions to the mean-square displacement of a molecule is presented. Concrete calculations are carried out for the H-bond subsystem of the two-dimensional model lattice water. The conjecture on the interrelation between the phase transition in the subsystem of H bonds and the glassification point is formulated. It is shown that (i) the glassification temperature is proportional to the H-bonding energy and (ii) the fragilities of glycerol-like liquids differ from each other as a consequence of distinct interaction energies between H bonds. The existence of a close connection between the fragility parameter and dielectric permittivity is established.