All-dielectric nanophotonics attracts more and more attention nowadays due to the possibility to control and configure light scattering on high-index semiconductor nanoparticles. It opens a room of opportunities for designing novel types of nanoscale elements and devices, paving the way to advanced technologies of light energy manipulation. One of the most pespective and interesting effects is directive light scattering provided by the so-called Kerker and anti-Kerker effects giving a possibility to realize Huygens light sources, fully transparent metasurfaces, different types of nanoantennae etc. Another one corresponds to the realization of so-called “anapole states” providing near-zero scattering accompanied with strong near-fields. Here we briefly review some new results on the induced invisibility regarding fully transparent metasurfaces based on the simultaneous cancellation of the forward and backward scattering via particular optical responses of multipoles (similar to Kerker effect), and invisible objects and structures governed by the novel type of anapoles – hybrid anapole states.