Recently emerged new platform for nanophotonics based on high-index dielectric nanoparticles utilizes optically-induced magnetic response via multipolar resonances, and provides many novel opportunities for subwavelength nonlinear optics. Here, we summarize our studies on nonlinear light scattering by high-permittivity dielectric nanoparticles and oligomers, and demonstrate approaches for achieving highly-efficient frequency conversion and directional harmonic radiation at the nanoscale. We analyze the multipolar nature of the generated electromagnetic fields by combining analytical and numerical methods. Our results delineate a roadmap towards design of miniature light sources and nonlinear photonic metadevices with exceptional characteristics.