<p>As a cluster overlap amplitude, the reduced-width amplitude is an important physical quantity for analyzing clustering in the nucleus depending on specified channels and has been calculated and widely applied in nuclear cluster physics. In this review, we briefly revisit the theoretical framework for calculating the reduced-width amplitude, as well as the outlines of cluster models to obtain microscopic or semi-microscopic cluster wave functions. We also introduce the recent progress related to cluster overlap amplitudes, including the implementation of cross-section estimation and extension to three-body clustering analysis. Comprehensive examples are provided to demonstrate the application of the reduced-width amplitude in analyzing clustering structures.</p>