  Many systems, including polymers and molecular liquids, when adequately
cooled and/or compressed, solidify into a disordered solid, i.e., a glass.
The~transition is not abrupt, featuring progressive decrease of the microscopic
mobility and huge slowing down of the relaxation.} A~distinctive aspect of
glass-forming materials is the microscopic dynamical heterogeneity (DH), i.e.,
the presence of regions with almost immobile particles coexisting with others
where highly mobile ones are located. Following the first compelling evidence
of a strong correlation between vibrational dynamics and ultraslow relaxation,
we posed the question if the vibrational dynamics encodes predictive
information on DH. Here, we review our results, drawn from molecular-dynamics
numerical simulation of polymeric and molecular glass-formers, with a special
focus on both the breakdown of the Stokes--Einstein relation between diffusion
and viscosity, and the size of the regions with correlated displacements.
