Laser cutting has emerged as a modern machining technology, garnering extensive application due to its benefits of high quality, precision, and efficiency. Cutting path planning is a pivotal step in the machining process, influencing both the quality of the cut and the time expenditure. This review synthesizes the advancements in path planning algorithms for laser cutting. Initially, the paper outlines the features of laser cutting technology and examines the associated constraints. Subsequently, a comprehensive review of the modeling and solution approaches for cutting path planning is provided. The paper concludes by comparing the strengths and limitations of various models and identifying avenues for future research in the field.