Laser Additive Manufacturing (LAM) presents unparalleled opportunities for fabricating complex, high-performance structures and components with unique material properties. Despite these advancements, achieving consistent part quality and process repeatability remains challenging. This paper provides a comprehensive review of various state-of-the-art in-situ process monitoring techniques, including optical-based monitoring, acoustic-based sensing, laser line scanning, and operando X-ray monitoring. These techniques are evaluated for their capabilities and limitations in detecting defects within Laser Powder Bed Fusion (LPBF) and Laser Directed Energy Deposition (LDED) processes. Furthermore, the review discusses emerging multisensor monitoring and machine learning (ML)-assisted defect detection methods, benchmarking ML models tailored for in-situ defect detection. The paper also discusses in-situ adaptive defect remediation strategies that advance LAM towards zero-defect autonomous operations, focusing on real-time closed-loop feedback control and defect correction methods. Research gaps such as the need for standardization, improved reliability and sensitivity, and decision-making strategies beyond early stopping are highlighted. Future directions are proposed, with an emphasis on multimodal sensor fusion for multiscale defect prediction and fault diagnosis, ultimately enabling self-adaptation in LAM processes. This paper aims to equip researchers and industry professionals with a holistic understanding of the current capabilities, limitations, and future directions in in-situ process monitoring and adaptive quality enhancement in LAM.