This paper reviews the development of the Analytic Hierarchy Process (AHP) since its inception. AHP is a multi-criteria decision-making method that has been widely used in various fields. The paper focuses on methodological developments rather than applications. Key areas of research in AHP include problem modelling, pairwise comparisons, judgement scales, derivation methods, consistency indices, incomplete matrices, synthesis of weights, sensitivity analysis, and group decisions. AHP uses a ratio scale for pairwise comparisons, allowing for relative values rather than absolute units. The 1-9 scale is based on psychological observations. The method involves creating a hierarchical structure of criteria, which helps in focusing on specific criteria and sub-criteria. The priority derivation involves methods such as the mean of the row, the geometric mean, and the eigenvalue method. The geometric mean is often preferred as it avoids rank reversal and provides more accurate results. Consistency checks are essential in AHP to ensure that the matrix used for comparisons is consistent. The consistency index (CI) and consistency ratio (CR) are used to measure and evaluate the consistency of the matrix. If the CR is less than 10%, the matrix is considered acceptable. Incomplete pairwise comparison matrices are also discussed, with methods to calculate missing comparisons and to handle incomplete data. The paper also addresses the issue of rank reversal, which occurs when the priorities of alternatives change due to the addition or removal of alternatives. This phenomenon is a significant debate in the AHP community. Group decision-making is another important aspect of AHP, where multiple experts' preferences are combined to reach a consensus. Different methods are used to aggregate preferences, including the geometric mean and weighted arithmetic mean methods. The paper also discusses the use of AHP in conjunction with other methods, such as mathematical programming techniques and fuzzy sets. Despite its widespread use, AHP still faces theoretical disputes, particularly regarding rank reversal and the assumption of criterion independence. The Analytic Network Process (ANP) is proposed as a potential solution to these issues. The paper concludes that AHP has reached a balance between model complexity and usability, making it a valuable tool for decision-making in various fields. Future developments may focus on improving the method's flexibility and adaptability to different decision-making scenarios.This paper reviews the development of the Analytic Hierarchy Process (AHP) since its inception. AHP is a multi-criteria decision-making method that has been widely used in various fields. The paper focuses on methodological developments rather than applications. Key areas of research in AHP include problem modelling, pairwise comparisons, judgement scales, derivation methods, consistency indices, incomplete matrices, synthesis of weights, sensitivity analysis, and group decisions. AHP uses a ratio scale for pairwise comparisons, allowing for relative values rather than absolute units. The 1-9 scale is based on psychological observations. The method involves creating a hierarchical structure of criteria, which helps in focusing on specific criteria and sub-criteria. The priority derivation involves methods such as the mean of the row, the geometric mean, and the eigenvalue method. The geometric mean is often preferred as it avoids rank reversal and provides more accurate results. Consistency checks are essential in AHP to ensure that the matrix used for comparisons is consistent. The consistency index (CI) and consistency ratio (CR) are used to measure and evaluate the consistency of the matrix. If the CR is less than 10%, the matrix is considered acceptable. Incomplete pairwise comparison matrices are also discussed, with methods to calculate missing comparisons and to handle incomplete data. The paper also addresses the issue of rank reversal, which occurs when the priorities of alternatives change due to the addition or removal of alternatives. This phenomenon is a significant debate in the AHP community. Group decision-making is another important aspect of AHP, where multiple experts' preferences are combined to reach a consensus. Different methods are used to aggregate preferences, including the geometric mean and weighted arithmetic mean methods. The paper also discusses the use of AHP in conjunction with other methods, such as mathematical programming techniques and fuzzy sets. Despite its widespread use, AHP still faces theoretical disputes, particularly regarding rank reversal and the assumption of criterion independence. The Analytic Network Process (ANP) is proposed as a potential solution to these issues. The paper concludes that AHP has reached a balance between model complexity and usability, making it a valuable tool for decision-making in various fields. Future developments may focus on improving the method's flexibility and adaptability to different decision-making scenarios.