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A review of recent theoretical and experimental results of the author in light scattering by high refractive index (HRI) particles is presented. It is shown, that the scattering is characterized by sharp, high-Q resonances affecting both the scattered field outside the particle and the one within it. However, the manifestation of the resonances in the former and later cases is quite different. For the scattered field outside the particle each partial mode exhibits an infinite cascade of the Fano resonances. Regarding the field within the particle, in contrast to the common believe that for a HRI particle this field is weak, it is shown that the resonances may give rise to a giant concentration of the electromagnetic field within the particle, exceeding the one in the incident wave in several orders of magnitude. These features of the problem provide physical grounds for numerous applications, especially those related to the design, engineering of high nonlinear heterogeneous nanostructures, and cloaking metamaterials.