Achieving precise kerf cut characteristics in laser beam cutting (LBC) is contingent upon the meticulous modeling of the process. This research delves into a comprehensive examination of laser beam cutting process models (LBCPM), with a specific focus on the material-centric perspective. The perception of LBCPM is pivotal in enhancing the performance of laser beam kerf, highlighting the pressing need for further investigation and review. Within this study, we embark on a comprehensive review of various types of LBCPM, encompassing the diverse characteristics of the kerf. These characteristics encompass geometrical, metallurgical, and mechanical facets. Our investigation categorizes LBC process modeling into three distinct dimensions: kerf characteristics, monitoring stage, and methodology. Theoretical models are expounded upon, leveraging mathematical equations, and physical material properties to provide a robust foundation. We delve into geometrical laser beam cutting process modeling, contrasting exact, numerical, regression, and machine learning methods for assessing top and bottom kerf width as well as kerf taper. The width of the heat-affected zone (HAZ), striation patterns including melt film thickness, and ablation depth will also be introduced, enriching our understanding of the process. Moreover, we explore the mechanical characteristics of kerf cut, presenting models for hardness and stress in the context of laser beam cutting. Additionally, we delve into surface quality, addressing roughness and dross models within the laser beam cutting process, providing insights into the factors that influence the final product's appearance and quality. In conclusion, this research not only offers a comprehensive overview of LBCPM but also identifies gaps in existing studies and sheds light on the future trends of laser beam cutting characteristics modeling. Through this multidimensional exploration, we aim to contribute to the continuous enhancement of laser beam cutting processes, ultimately advancing the field's understanding and application.