If liquids, polymers, bio-materials, metals and molten salts can avoid crystallization during cooling or compression, they freeze into a microscopically disordered solid-like state, a glass. On approaching the glass transition, particles become trapped in transient cagesin which they rattle on picosecond timescalesformed by their nearest neighbours; the particles spend increasing amounts of time in their cages as the average escape time, or structural relaxation time α, increases from a few picoseconds to thousands of seconds through the transition. Owing to the huge difference between relaxation and vibrational timescales, theoretical studies addressing the underlying rattling process have challenged our understanding of the structural relaxation. Numerical and experimental studies on liquids and glasses support the theories, but not without controversies (for a review see ref.21). Here we show computer simulations that, when compared with experiments, reveal the universal correlation of the structural relaxation time (as well as the viscosity ) and the rattling amplitude from glassy to low-viscosity states. According to the emerging picture the glass softens when the rattling amplitude exceeds a critical value, in agreement with the Lindemann criterion for the melting of crystalline solids and the free-volume model. © 2008 Nature Publishing Group.