This review overviews the state of the art of research into high-index dielectric nanoresonators and their use in functional photonic nanostructures at optical frequencies. We start by providing the motivations for this research area and by putting it into context with the more well-established subfields of nanophotonics, in particular nanoplasmonics. Following the introduction, fundamental concepts regarding the optical properties of subwavelength dielectric nanoresonators are established. To this end, we provide a brief summary of the Mie theory, before focussing on optically induced magnetic response in Mie-resonant dielectric nanoparticles. We discuss the influence of the nanoparticle’s shape on its optical response, and provide an overview of directional effects that can occur when light is scattered by a Mie-resonant nanoparticle. We then dedicate a few words to technology-related aspects, including an overview of fabrication methods for Mie-resonant dielectric nanoparticles. Next, recent progress on all-dielectric nanoantennas is presented, focussing on strategies to locally enhance optical near-fields and to achieve directional emission patterns. We then turn to all-dielectric metasurfaces and their potential applications. We touch on dielectric metamaterial reflectors and Fano-resonant dielectric metasurfaces, before discussing graded Mie-resonant dielectric metasurfaces for wavefront control applications in more detail. Following this, an overview of the recent progress in active, tunable and nonlinear dielectric nanostructures is provided. Finally, prospects and challenges are discussed, particularly the realization of highly efficient Mie-resonant nanostructures at visible frequencies, the integration of Mie-resonant nanostructures with active and functional materials, and the construction of three-dimensional high-index dielectric nanostructures.