We summarise the recent advances on the generation of high-order harmonics in optical metaphotonic structures such as isolated resonators and metasurfaces. In such subwavelength-patterned structures, extreme nonlinear effects are expected from the interaction between ultrafast laser pulses and structured planar surfaces which support various resonances. High-harmonic generation (HHG) is considered as an effective tool for realising extreme ultraviolet light sources and attosecond pulses, and it was observed previously in gases, liquids, and solids. Resonant metaphotonics can offer a novel sub-wavelength platform for the HHG effects, and it can provide new strategies for the design of efficient integrated light sources. We start our discussion from a brief overview of HHG in gases, liquids, and unstructured solids, and then move to summarising the recent experimental observations of HHG in individual resonant nanoparticles and resonant dielectric metasurfaces. We focus on different types of resonances (plasmonic vs. Mie resonances vs. bound states in the continuum), and also present the observation of non-integer power dependencies of the generated harmonics driven by strong resonances. We also mention current unsolved problems and identify new promising research directions involving HHG in metasurfaces.