Nonradiative sources of light such as anapoles and bound states in the continuum (BICs) were initially proposed in quantum mechanics and astrophysics, and they did not attract much attention in photonics for a long time. However, primarily due to the rapid development of metamaterials and metaphotonics, it was recognized that such states are very prospective for efficient trapping of light, amplification of local fields, control of scattering, and also nonlinear manipulation of light at the nanoscale. Metaphotonics provides a broad variety of resonant dielectric structures, including optical gratings, metasurfaces, photonic crystals, and single resonators for a precise engineering of high values of quality factor (Q-factor) of the resonant states and their optical response. In the last ten years, nonradiating states matured from pure conceptual fundamental works to experimental demonstrations and original applications in photonics and radiophysics. They promised functional tools for controlling electromagnetic radiation of different spectral ranges from visible light to microwaves.