Free from phase-matching constraints, plasmonic metasurfaces have contributed significantly to the control of optical nonlinearity and enhancement of nonlinear generation efficiency by engineering subwavelength meta-atoms. However, high dissipative losses and inevitable thermal heating limit their applicability in nonlinear nanophotonics. All-dielectric metasurfaces, supporting both electric and magnetic Mie-type resonances in their nanostructures, have appeared as a promising alternative to nonlinear plasmonics. High-index dielectric nanostructures, allowing additional magnetic resonances, can induce magnetic nonlinear effects, which, along with electric nonlinearities, increase the nonlinear conversion efficiency. In addition, low dissipative losses and high damage thresholds provide an extra degree of freedom for operating at high pump intensities, resulting in a considerable enhancement of the nonlinear processes. We discuss the current state of the art in the intensely developing area of all-dielectric nonlinear nanostructures and metasurfaces, including the role of Mie modes, Fano resonances, and anapole moments for harmonic generation, wave mixing, and ultrafast optical switching. Furthermore, we review the recent progress in the nonlinear phase and wavefront control using all-dielectric metasurfaces. We discuss techniques to realize all-dielectric metasurfaces for multifunctional applications and generation of second-order nonlinear processes from complementary metal–oxide–semiconductor-compatible materials.