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Polyacrylonitrile (PAN) is a versatile man-made polymer and has been used in a large array of products since its first mass production in the mid 40s. Among all applications of PAN the widely used application is in manufacture of precursor fiber for fabrication of carbon fibers. The process of PAN-based carbon fiber production comprises fiber spinning, thermal stabilization and carbonization stages. Carbon fiber properties are significantly dependent on the quality of PAN precursor fiber and in particular the process parameters involved in thermal stabilization. This paper is the first comprehensive review that provides a general understanding of the links between PAN fiber structure, properties, and its stabilization process along with the use of mathematical modelling as a powerful tool in prediction and optimization of the processes involved. Since the promise of the mathematical modelling is to predict the future behaviour of the system and the value of the variables for the unseen or unmeasured domain of variables; and in the era of industry 4.0 rise, this review will be valuable in further understanding of the intricate processes of carbon fiber manufacture and utilising the advanced mathematical modelling using machine learning techniques to predict and optimize a range of critical factors that control the quality of PAN and resultant carbon fibers. [ABSTRACT FROM AUTHOR]