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Over the past 30 years or so, desingularized boundary integral equations (DBIEs) have been used to study water wave dynamics and body motion dynamics. Within the potential flow modeling, unlike conventional boundary integral methods, a DBIE separates the integration surface and the control (collocation) surface, resulting in a BIE with non-singular kernels. The desingularization allows simpler and faster numerical evaluation of the boundary integrals, and consequently faster numerical solutions. In this paper, derivations of different forms of DBIEs are given and the fundamental aspects and advantages of the DBIEs are reviewed and discussed. Numerical examples of applications of DBIEs in wave dynamics and body motion dynamics are given and the outlook of future development of the desingularized methods is discussed