  Freed from phase-matching constraints, plasmonic metasurfaces have
contributed significantly to the control of the optical nonlinearity and
enhancing the nonlinear generation efficiency by engineering subwavelength
meta-atoms. However, the high dissipative losses and the 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. In this review, 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 CMOS
compatible materials.
