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Frequency conversion processes, such as second- and third-harmonic generation, are commonly realized in nonlinear optics, offering opportunities for applications in photonics, chemistry, material science and biosensing. Given the inherently weak nonlinear response of natural materials, optically large samples and complex phase-matching techniques are typically required to realize significant nonlinear responses. To produce similar effects in much smaller volumes, current research has been devoted to the quest of synthesizing novel materials with enhanced optical nonlinearities at moderate input intensities. In particular, several approaches to engineer 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 based on plasmonic and high-index dielectric resonators, as well as semiconductor-loaded plasmonic metasurfaces. In this context, we also discuss recent advances in controlling the optical wavefront of generated nonlinear waves using metasurfaces. Finally, we compare viable approaches to enhance nonlinearities in ultrathin metasurfaces, and we offer an outlook on the future development of this exciting field of research. [ABSTRACT FROM AUTHOR]