This paper surveys methods for detecting abrupt changes (such as failures) in stochastic dynamical systems. It focuses on linear systems, though the basic concepts apply to other system classes. The methods range from designing failure-sensitive filters to statistical tests on filter innovations and jump process formulations. Trade-offs between complexity and performance are discussed. The paper discusses the problem of detecting abrupt changes in dynamical systems, such as actuator and sensor failures in aircraft control and arrhythmias in electrocardiography. It emphasizes the need for systems that can detect these changes quickly while minimizing performance degradation during normal operation. The paper reviews various failure detection methods, including failure-sensitive filters, voting systems, multiple hypothesis filters, and jump process formulations. It also discusses the use of statistical tests, such as chi-squared tests, for detecting anomalies in filter innovations. The paper highlights the importance of considering system redundancy, computational complexity, and the trade-offs between these factors in designing effective failure detection systems. It also discusses the use of probabilistic models and likelihood ratio methods for failure detection, including the generalized likelihood ratio (GLR) approach. The GLR method is particularly effective in detecting various types of failures, including dynamics jumps, dynamic steps, sensor jumps, and sensor steps. The paper concludes that while failure detection systems can be complex, they are essential for ensuring the reliability and safety of dynamic systems.This paper surveys methods for detecting abrupt changes (such as failures) in stochastic dynamical systems. It focuses on linear systems, though the basic concepts apply to other system classes. The methods range from designing failure-sensitive filters to statistical tests on filter innovations and jump process formulations. Trade-offs between complexity and performance are discussed. The paper discusses the problem of detecting abrupt changes in dynamical systems, such as actuator and sensor failures in aircraft control and arrhythmias in electrocardiography. It emphasizes the need for systems that can detect these changes quickly while minimizing performance degradation during normal operation. The paper reviews various failure detection methods, including failure-sensitive filters, voting systems, multiple hypothesis filters, and jump process formulations. It also discusses the use of statistical tests, such as chi-squared tests, for detecting anomalies in filter innovations. The paper highlights the importance of considering system redundancy, computational complexity, and the trade-offs between these factors in designing effective failure detection systems. It also discusses the use of probabilistic models and likelihood ratio methods for failure detection, including the generalized likelihood ratio (GLR) approach. The GLR method is particularly effective in detecting various types of failures, including dynamics jumps, dynamic steps, sensor jumps, and sensor steps. The paper concludes that while failure detection systems can be complex, they are essential for ensuring the reliability and safety of dynamic systems.