Melting is analyzed dynamically as a problem of localization at a liquid-solid interface. A Lindemann-like criterion of melting is derived in terms of particular vibrational amplitudes, which turn out to equal a universal quotient (about one-tenth) of the molecular spacing, at the interface. The near universality of the Lindemann ratio apparently arises owing to strongly overdamped dynamics near melting, and despite the anharmonic interactions being system-specific. A similar criterion is derived for structural displacements in the bulk of the solid, in particular the premelted layer; the criterion is no longer strictly universal, but still depends only on the harmonic properties of the solid. We further compute the dependence of the magnitude of the elemental molecular translations, in deeply supercooled fluids, on the temperature and the high frequency elastic constants. We show explicitly that the surface tension between distinct liquid states, near the glass transition of a supercooled liquid, is nearly evenly split between entropic and energetic contributions.