This paper surveys methods for detecting abrupt changes, or failures, in stochastic dynamical systems, focusing primarily on linear systems. The methods discussed include the design of specific failure-sensitive filters, statistical tests on filter innovations, and jump process formulations. The paper emphasizes the trade-offs between system complexity and performance, such as the balance between rapid response to failures and tolerance of noise and false alarms. The author reviews various techniques, including "failure-sensitive" filters, voting systems, multiple hypothesis filter-detectors, and innovations-based detection systems, each with its own advantages and limitations. The generalized likelihood ratio (GLR) method, which can handle a wide range of failure modes, is highlighted for its ability to provide both alarm and isolation information, though it comes with significant computational complexity. The paper also discusses the practical implementation and performance of these methods, providing insights into their effectiveness in different applications.This paper surveys methods for detecting abrupt changes, or failures, in stochastic dynamical systems, focusing primarily on linear systems. The methods discussed include the design of specific failure-sensitive filters, statistical tests on filter innovations, and jump process formulations. The paper emphasizes the trade-offs between system complexity and performance, such as the balance between rapid response to failures and tolerance of noise and false alarms. The author reviews various techniques, including "failure-sensitive" filters, voting systems, multiple hypothesis filter-detectors, and innovations-based detection systems, each with its own advantages and limitations. The generalized likelihood ratio (GLR) method, which can handle a wide range of failure modes, is highlighted for its ability to provide both alarm and isolation information, though it comes with significant computational complexity. The paper also discusses the practical implementation and performance of these methods, providing insights into their effectiveness in different applications.