AbstractProducing materials with a desired refraction coefficient is of great theoreti- cal and practical interest. There was no general method for creating such materials, except the method, developed by the author. It was not even known that such a method do ex- ist. The theoretical basis of this method is the asymptotic solution of the many-body wave scattering problem for many small bodies with prescribed boundary impedances. Multiple scattering is essential in our theory. The small bodies are embedded in a bounded region D, filled with a material with a known refraction coefficient n0(x). Our basic physical assumption is a ≪ d ≪ λ, where a is the characteristic size of the small particle, d is the minimal distance between neighboring particles, and λ is the wave length in D. The asymptotic of the solution to the above many-body scattering problem is derived for a → 0.