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Paris 12, 61 ave. du General de Gaulle, 94040 Creteil Cedex; Laboratoire d'Analyse Numerique, Tour 55-65-5* me etage, Universite Paris 6,4 place Jussieu, 75252 Paris Cedex 05, France (MS received 29 June 1994}This paper generalises the notion of two-scale convergence to the case of multiple separated scales of periodic oscillations. It allows us to introduce a multi-scale convergence method for the reiterated homogenisation of partia/ differentia/ equations wilh oscillating coefficients. This new method is applied to a model problem with a finite or infinite number of microscopic scales, namely the homogenisation of the heat equation in a composite material. Finally, it is generalised to handle the homogenisation of the Neumann problem in a perforated domain.