Nonlinear optics is a well-established field of research that traditionally relies on the interaction of light with macroscopic nonlinear media over distances significantly greater than the wavelength of light. However, the recently emerged field of optical metasurfaces provides a novel platform for studying nonlinear phenomena in planar geometries. Nonlinear optical metasurfaces introduce new functionalities to the field of nonlinear optics extending them beyond perturbative regimes of harmonic generation and parametric frequency conversion, being driven by mode-matching, resonances, and relaxed phase-matching conditions. Here we review the very recent advances in the rapidly developing field of nonlinear metasurface photonics, emphasizing multi-frequency and cascading effects, asymmetric and chiral frequency conversion, nonperturbative nonlinear regimes, and nonlinear quantum photonics, empowered by the physics of Mie resonances and optical bound states in the continuum.