In this Article, we review a novel, rapidly developing field of modern light science named alldielectric nanophotonics. This branch of nanophotonics is based on the properties of high-index dielectric nanoparticles which allow for controlling both magnetic and electric responses of a nanostructured matter. Here, we discuss optical properties of high-index dielectric nanoparticles, methods of their fabrication, and recent advances in practical applications, including the quantum source emission engineering, Fano resonances in all-dielectric nanoclusters, surface enhanced spectroscopy and sensing, coupled-resonator optical waveguides, metamaterials and metasurfaces, and nonlinear nanophotonics.