This document reviews research on optimizing parameters for laser beam machining (LBM) using Taguchi and grey relational analysis methods. It summarizes several past studies that aimed to optimize LBM quality characteristics like kerf width, surface roughness, heat affected zone, etc. by adjusting parameters like laser power, cutting speed, gas pressure, focal position. The document outlines the Taguchi method approach of using orthogonal arrays and signal-to-noise ratios to identify optimal parameter settings. It also describes grey relational analysis and its steps of preprocessing data, calculating grey relational coefficients and grades, and analyzing results to select optimal parameters. Tables show examples of optimized cutting speeds for different materials and thicknesses from previous works.