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This paper presents a summary of various localized collocation schemes and their engineering applications. The basic concepts of localized collocation methods (LCMs) are first introduced, such as approximation theory, semianalytical collocation methods and localization strategies. Based on these basic concepts, five different formulations of localized collocation methods are introduced, including the localized radial basis function collocation method (LRBFCM) and the generalized finite difference method (GFDM), the localized method of fundamental solutions (LMFS), the localized radial Trefftz collocation method (LRTCM), and the localized collocation Trefftz method (LCTM). Then, several additional schemes, such as the generalized reciprocity method, Laplace and Fourier transformations, and Krylov deferred correction, are introduced to enable the application of the LCM to large-scale engineering and …