  Nonradiating sources of energy have traditionally been studied in quantum
mechanics and astrophysics, while receiving a very little attention in the
photonics community. This situation has changed recently due to a number of
pioneering theoretical studies and remarkable experimental demonstrations of
the exotic states of light in dielectric resonant photonic structures and
metasurfaces, with the possibility to localize efficiently the electromagnetic
fields of high intensities within small volumes of matter. These recent
advances underpin novel concepts in nanophotonics, and provide a promising
pathway to overcome the problem of losses usually associated with metals and
plasmonic materials for the efficient control of the light-matter interaction
at the nanoscale. This review paper provides the general background and several
snapshots of the recent results in this young yet prominent research field,
focusing on two types of nonradiating states of light that both have been
recently at the center of many studies in all-dielectric resonant meta-optics
and metasurfaces: optical {\em anapoles} and photonic {\em bound states in the
continuum}. We discuss a brief history of these states in optics, their
underlying physics and manifestations, and also emphasize their differences and
similarities. We also review some applications of such novel photonic states in
both linear and nonlinear optics for the nanoscale field enhancement, a design
of novel dielectric structures with high-$Q$ resonances, nonlinear wave mixing
and enhanced harmonic generation, as well as advanced concepts for lasing and
optical neural networks.

Optics