  Asymptotic solution to many-body wave scattering problem is given in the case
of many small scatterers. The small scatterers can be particles whose physical
properties are described by the boundary impedances, or they can be small
inhomogeneities, whose physical properties are described by their refraction
coefficients. Equations for the effective field in the limiting medium are
derived. The limit is considered as the size $a$ of the particles or
inhomogeneities tends to zero while their number $M(a)$ tends to infinity.
These results are applied to the problem of creating materials with a desired
refraction coefficient. For example, the refraction coefficient may have
wave-focusing property, or it may have negative refraction, i.e., the group
velocity may be directed opposite to the phase velocity. This paper is a review
of the author's results presented in MR2442305 (2009g:78016), MR2354140
(2008g:82123), MR2317263 (2008a:35040), MR2362884 (2008j:78010), and contains
new results.
