This review explores the application of Greybox Modelling, specifically Physics-Informed Machine Learning (PIML), in the context of Structural Health Monitoring (SHM) within civil engineering. SHM is vital for ensuring the safety and durability of infrastructure namely bridges. Greybox Modelling, merging physics-based principles with data-driven techniques, offers a functional approach for understanding complex structural behaviors. By leveraging existing knowledge of physical laws, this method extracts valuable insights from sensor data. The integration of Greybox Modelling takes SHM a step further. By applying machine learning algorithms with the physics that govern civil engineering structures, Greybox Modelling bridges the gap between data-driven models and physics-based models. This leads more accurate predictions and interpretable outcomes. This review discusses findings from literature, recent studies, that illustrate the effectiveness of Greybox Modelling, specifically PIML in various civil engineering applications. These case studies highlight how this approach can adapt to different structural designs and environmental conditions typically encountered in civil engineering. Moreover, they indicate its potential for damage detection and early warning systems, which enhance the safety and efficiency of infrastructure. The paper also addresses current challenges and outlining future directions in the field.