Glacier surges are episodes of strongly accelerated glacier flow. The phenomenon is important to study not only for the advancement of geosciences, but also for risk management and human safety. Unfortunately, surge events remain difficult to observe and predict due to their short duration relative to the entire surge cycle, as well as to their spatial heterogeneity and diverse manifestations. Over the past decades, satellite remote sensing has become the primary source of information for monitoring surge activity, with synthetic aperture radar (SAR) offering unique advantages in terms of temporal coverage and independence from illumination and weather conditions. This article provides an overview of SAR-based methods applied to the study of glacier surges, including interferometry, offset tracking, altimetry, backscatter and coherence map analysis. We discuss how individual SAR products detect complementary aspects of surge dynamics, such as velocity increase, elevation change, terminus fluctuations, surface crevassing and the propagation of instability front. Based on the reviewed literature, we identify current research trends ranging from surge detection and glacier classification to long-term monitoring of individual glaciers, and we point to the main challenges that arise from the characteristics of surge behavior. Finally, we discuss directions for future research, including the integration of multi-aspect SAR observations, the use of machine learning techniques, and the potential role of new SAR methods in early detection and prediction of surge events.