Sustainable Aviation Fuel (SAF) is increasingly recognized as a vital solution for reducing the carbon footprint of the aviation industry. However, its production and supply chain management involve significant complexities that demand advanced modeling and optimization strategies. Despite the rising interest in SAF, the existing supply chain infrastructure remains immature, with critical barriers such as limited feedstock availability, production scalability challenges and cost competitiveness impeding broader adoption. This study systematically reviews recent advancements in SAF supply chain modeling and optimization, highlighting key methodologies such as mathematical programming, machine learning and multi-objective optimization for enhancing efficiency and sustainability. Some challenges, such as high production costs, complexities of multi-stage processing, regulatory inconsistencies, feedstock availability and variability, as well as research gaps, have been identified. This review also covers mitigation strategies to support the establishment of SAF as a scalable and viable alternative to conventional aviation fuel.