With use of universal representation Green's tensors for composite spherical bodies, we obtained the solution for number of actual problems of electromagnetic excitation and diffraction. In [1] results of research of influence of model of a head of the person on characteristics of radiation of the antenna of the mobile phone are given. In [2] shielding disseminating properties of a thin semiconducting shell were investigated. In the given report the basic attention is given to investigation of diffraction on spherical bodies from materials with double negative index of refraction DNGM (left-handed materials LHM). More than in thirty years after publication of pioneer article of Veselago V.G. [3] the flow of the theoretical and experimental works devoted to research of the physical phenomena in new materials and opportunities of their realization [4], [5], [6] has appeared. Deflections from a number of the fundamental physical laws were revealed: Cherenkov's radiations, Doppler's effect, Snell's laws, etc. The reports devoted to research of propagation of waves in LHM, to their electromagnetic excitation have appeared. We obtain the extensive data concerning diffraction of electromagnetic waves on spherical structures from DNGM. A part of results of these researches are present in this report.