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 the designing novel types of nanoscale elements and devices, and paves a way to advanced technologies of light energy manipulation. One of the most perspective and interesting effects is directive light scattering provided by the so-called Kerker and anti-Kerker effects giving a possibility to realize Huygens source of light, fully transparent metasurfaces, different types of nanoatennae 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.