  I review the four mechanisms of bound states in the continuum (BICs) in
application to microwave and acoustic cavities open to directional waveguides.
The most simple are the symmetry protected BICs which are localized inside the
cavity because of the orthogonality of the eigenmodes to the propagating modes
of waveguides. However, the most general and interesting is the
Friedrich-Wintgen mechanism when the BICs are result of full destructive
interference of outgoing resonant modes. The third type of the BICs, the
Fabry-Perot BICs, occur in a double resonator system when each resonator can
serve as an ideal mirror. At last, the accidental BICs can be realized in the
open cavities with no symmetry like the open Sinai billiard in which the
eigenmode of the resonator can become orthogonal to the continuum of the
waveguide accidentally by a smooth deformation of the eigenmode. We also review
the one-dimensional systems in which the BICs occur owing to full destructive
interference of two waves separated by spin or polarization or by paths in the
Aharonov-Bohm rings. We widely use the method of effective non-Hermitian
Hamiltonian equivalent to the coupled mode theory which detects bound states in
the continuum (BICs) by finding zero widths resonances.
