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1.Reaction tunnelling rates This talk will explain how semi-classical instanton theory can be formulated very simply in terms of ring-polymers1-4,to yield the dominant ‘instanton’ tunneling path through a potential barrier.Harmonic fluctuations around this path yield an expression for the instanton tunneling rate,which can be shown3 to be identical to the instanton rate originally derived by Miller5 from the exact flux-side time-correlation function.The approach also explains why the simulation technique of Ring Polymer Molecular Dynamics (RPMD)6 gives realistic predictions of quantum rates in the deep-tunnelling regime.