Safety and reliability of large critical infrastructure such as long-span bridges, high-rise buildings, nuclear power plants, high-voltage transmission towers, rotating machinery, and so on, are important for a modern society. Research on reliability and safety analysis started with a “small data” problem dealing with relative scarce lifetime or failure data. Later, degradation modeling that uses performance deterioration, or, condition data collected from in-service inspections or online health monitoring became an important 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 lives. The effect of these changes on the degradation modeling, prognosis, and safety management is 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 are classified into four categories: physics-based, knowledge-based, data-driven, and hybrid approaches.