Low-voltage distribution networks (LVDNs) serve as the final delivery layer of the electricity system, directly influencing reliability, public safety, customer service quality, and the integration of distributed energy resources. Despite their importance, LVDNs have historically received less monitoring than transmission and medium-voltage systems due to their scale, cost, and deployment complexity. Non-contact magnetic sensing has emerged as a promising alternative to invasive measurement methods for these networks. Among magnetic sensor types, giant magnetoresistive (GMR) devices are appealing because they offer high sensitivity, compactness, low power consumption, and compatibility with embedded electronics. This review assesses the current state of GMR-based monitoring for overhead and low-voltage applications, focusing on non-contact current measurement, fault detection, and fault classification. It first examines the operating characteristics of LVDNs and the unique challenge of detecting low- and high-impedance faults. Next, it outlines the physical principles behind GMR sensing, compares GMR with Hall, AMR, TMR, current transformer, and Rogowski-coil technologies, and discusses the use of multi-axis sensor heads to address cross-coupled magnetic fields in three-phase setups. Special focus is given to calibration, alignment, temperature effects, electromagnetic interference, packaging, wireless deployment, and data-driven classification. The review concludes that GMR sensors are well-suited for scalable, non-contact monitoring, but widespread adoption in the field will require better low-voltage fault datasets, standardized calibration procedures, long-term environmental testing, and closer integration with digital-twin and smart-meter infrastructures.