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I review the four mechanisms of bound states in the continuum (BICs) in the application of microwave and acoustic cavities open to directional waveguides. The most simple are 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 the result of the fully destructive interference of outgoing resonant modes. The third type of BICs, Fabry–Perot BICs, occurs in a double resonator system when each resonator can serve as an ideal mirror. Finally, 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 due to a smooth deformation of the eigenmode. We also review the one-dimensional systems in which the BICs occur owing to the fully destructive interference of two waves separated by spin or polarization or by paths in the Aharonov–Bohm rings. We make broad use of the method of effective non-Hermitian Hamiltonian equivalent to the coupled mode theory, which detects BICs by finding zero-width resonances. [ABSTRACT FROM AUTHOR]