Bound states in the continuum are an excellent example of how a solution to a simple problem of quantum mechanics, obtained about a century ago, can serve as an incentive to study the whole spectrum of resonance phenomena in wave physics. Due to the giant radiative lifetime, bound states in a continuum have found many applications in various fields of physics, in particular, in hydrodynamics, atomic physics, and acoustics. In this review, we present a broad view to the physics of bound states in the continuum and related effects, focusing primarily on photonic dielectric structures. We consider the history of the development of this direction, the main physical mechanisms of the formation of bound states in the continuum, specific examples of structures that can support such states. We also discuss their possible practical applications in optics, photonics, and radiophysics.