  Frequency conversion processes, such as second- and third-harmonic
generation, are one of the most common effects in nonlinear optics which offer
many opportunities for photonics, chemistry, material science,
characterization, and biosensing. Given the inherently weak nonlinear response
of natural materials, one typically relies on optically large samples and
complex phase-matching techniques to achieve nonlinear effects. A direct
translation of these approaches to small dimensions, however, is extremely
challenging, because nonlinear effects locally comparable to the linear
response cannot be induced without reaching extremely high light intensities
leading to the material breakdown. For this reason, the quest to synthesize
novel materials with enhanced optical nonlinearities at moderate input
intensities is very active nowadays. In the last decade, several approaches to
engineering the nonlinear properties of artificial materials, metamaterials,
and metasurfaces have been introduced. Here, we review the current state of the
art in the field of small-scale nonlinear optics, with special emphasis on
high-harmonic generation from ultrathin metasurfaces, including those based on
plasmonic and high-index dielectric resonators, as well as semiconductor-loaded
plasmonic metasurfaces. We discuss the role of specific electromagnetic field
configurations for efficient harmonic generation, including magnetic dipole,
Fano resonance, and anapole ones. We also discuss the most recent advances in
controlling the phase front profiles of generated nonlinear waves enabled by
nonlinear metasurfaces consisting of nanoresonators with judiciously tailored
shapes. Finally, we compare viable approaches to enhance nonlinearities without
phase matching constraints in ultrathin metasurfaces, and offer a perspective
and outlook on the future development of this exciting field.
