Rapid progress in plasmonics is driven by the ability to enhance near-field effects with subwavelength localization of light. Recently, we observe the emergence of a new branch of nanophotonics aiming at the manipulation of strong optically-induced electric and magnetic Mie-type resonances in dielectric and semiconductor nanostructures with high refractive index. Unique advantages of dielectric resonant optical nanostructures over their metallic counterparts are low dissipative losses, low heating, and the enhancement of both electric and magnetic fields. In this talk, I will review this new emerging field of nanophotonics and metasurfaces and demonstrate that Mie-type resonances in high-index dielectric nanoparticles and subwavelength structures can be exploited for new physics and novel functionalities of photonic structures especially in the nonlinear regime.