Safety and reliability of large critical infrastructure systems such as long-span bridges, high-rise buildings, nuclear power plants, high-voltage transmission towers, rolling-element bearing and so on are important for a modern society. Research on reliability and safety analysis started with a `small data' business dealing with relative scarce lifetime or failure data. Later, degradation modelling that uses performance deterioration or condition data collected from inservice inspections or online health monitoring became an important analytical tool for reliability prediction and maintenance planning of highly reliable engineering systems. Over the past decades, a large number of degradation models have been developed to characterize and quantify the underlying degradation mechanism using direct and indirect measurements. Recent advancements in artificial intelligence, remote sensing, big data analytics, and Internet of Things are making far-reaching impacts on almost every aspect of our life. How these changes will affect the degradation modelling, health prognosis, and safety management is an interesting questions to explore. This paper presents a comprehensive, forward-looking review of the various degradation models and their practical applications to damage prognosis and management of critical infrastructure. The degradation models were classified into four categories: physics-based, knowledge-based, data-driven, and hybrid approaches.