Analytic Hierarchy Process (AHP) is a widely applied technique in multi-criteria decision-making (MCDM) problems. Over time, numerous hybridizations, improvements, and modifications have been proposed to address the shortcomings of traditional AHP. Considering the sheer number of the AHP-based methods, scholars/practitioners are faced with certain challenges when selecting a suitable method due to: (i) lack of adequate knowledge on pros and cons of different AHP approaches, (ii) difficulties and limitations in the application and analysis, and (iii) uncertainties about the suitability of the method. As a result, there is a need for a comprehensive review functioning as a guidance when choosing the best-suited approach considering the specific features of the problem at hand. This paper, therefore, reviews articles published between 2010 and 2023 that have proposed a hybrid, improved, or modified AHP and classifies them based on three main categories of contributions: (A) consistency improvements, (B) reducing the difficulties or limitations, and (C) increasing the accuracy of the results. These categories are further discussed based on the nature of variation (hybridizing with fuzzy sets, metaheuristic algorithms, modification of AHP structure, and hybridization with other approaches). A comprehensive summary table is provided to showcase the strengths and weaknesses of each method, and a roadmap is put forward for scholars and industry experts assisting them in the selection of the appropriate method considering various aspects of problems. Finally, directions for future research are discussed.