Abstract: The presentation focuses on statistical time series based damage and fault diagnosis for structures and engineering systems operating under uncertainty. The various versions of the problem formulation are reviewed, and a concise, yet critical, overview of the main principles, underlying assumptions, and available approaches is presented. The need for robustness, arising from the necessity for counteracting the effects of uncertain Environmental and Operational Conditions (EOCs), but also those associated with populations of similar structures and systems, is demonstrated. The main concepts and approaches of robust methods are then critically reviewed, with emphasis on conceptual and practical simplicity, ease of use, operation with a low number of sensors and limited numbers of training signals, physical interpretability, and the achievement of high-performance levels for even `minor’ fault levels. The novel and holistic Functional Model (FM) based method, within which the subproblems of damage/fault detection, precise localization, and level estimation may be seamlessly addressed, is subsequently introduced. Its various forms, including those based on measurable EOCs and recent ones capable of eliminating this requirement, are discussed. Application case studies, pertaining to damage diagnosis for engineering structures of various types and on-board fault diagnosis for railway suspension systems under uncertainty, are then presented, with diagnostic performance systematically assessed via Receiver Operating Characteristic curves and related metrics. The presentation concludes with remarks on current achievements, the technology’s status and limitations, and perspectives on the way forward.