Understanding the extent of urban flooding is crucial for assessing building damage, casualties and economic losses. Synthetic Aperture Radar (SAR) technology offers significant advantages for mapping flooded urban areas due to its ability to collect data regardless weather and solar illumination conditions. However, the wide range of existing methods makes it difficult to choose the best approach for a specific situation and to identify future research directions. Therefore, this study provides a comprehensive review of current research on urban flood mapping using SAR data, summarizing key characteristics of floodwater in SAR images and outlining various approaches from scientific articles. Additionally, we provide a brief overview of the advantages and disadvantages of each method category, along with guidance on selecting the most suitable approach for different scenarios. This study focuses on the challenges and advancements in SAR-based urban flood mapping. It specifically addresses the limitations of spatial and temporal resolution in SAR data and discusses the essential pre-processing steps. Moreover, the article explores the potential benefits of Polarimetric SAR (PolSAR) techniques and uncertainty analysis for future research. Furthermore, it highlights a lack of open-access SAR datasets for urban flood mapping, hindering development in advanced deep learning-based methods. Besides, we evaluated the Technology Readiness Levels (TRLs) of urban flood mapping techniques to identify challenges and future research areas. Finally, the study explores the practical applications of SAR-based urban flood mapping in both the private and public sectors and provides a comprehensive overview of the benefits and potential impact of these methods.