Laser directed energy deposition (LDED) is an emerging branch of metal-based additive manufacturing (AM) processes, offering unprecedented capabilities for high-performance fabrication with complex geometries and near-net shapes. This technology is gathering increasing attention from industries such as biomedical, automotive, and aerospace. However, achieving consistent part quality and desired material properties is challenging due to intricate processing parameters and potential process defects such as dynamic melt-pool behavior and localized heat accumulation. This paper reviews recent advances in on-line quality control, focusing on in-situ measurement and closed-loop control for efficient assurance of LDED-fabricated parts. The quality principles, encompassing accuracy and material performance, are summarized to lay a foundation for understanding the mechanisms of quality defects and influencing factors. This review explores and thoroughly compares advancements in indirect process measurements, such as optical, thermal, and acoustic monitoring with direct quality measurements, including laser-line scanning and operando synchrotron X-ray imaging. Depending on the sensing techniques, this paper highlights a hierarchical control strategy for adaptive parameter regulation on intra-layer and inter-layer scales. The requirements and performance of various state-of-the-art controllers are critically compared to indicate their suitable applications. The importance of machine learning in detecting process anomalies and predicting build quality based on sensory signals is also outlined. Future directions are proposed towards adaptive, automated, and intelligent quality control, with a focus on multi-modal monitoring, physics-informed neural networks for interpretable analysis, and multi-objective control applications.