An important aspect of network management is fault management-determining, locating, isolating and correcting faults in the network. The paper deals with the algorithms for detecting faults, i.e., behavior of the network different from specifications. It is important for communication networks to detect faults "in-process" i.e., while the network is in its normal operation. Thus, we detect faults by examining the input-output behavior without forcing the system to specialized inputs explicitly for testing. Such testing is commonly called passive testing. We model the network as a finite state machine and develop procedures for passive testing including the required data structure, efficient implementations and the complexity of our procedures. We start with fully observable and deterministic machines and then study more realistic models: partially observable and nondeterministic machines. We also discuss extensions to communicating finite state machines and machines extended with parameters and variables. We apply our techniques to management of a signaling network operating under the Signaling System 7 (SS7) and report experimental results, which show the feasibility of applying passive testing to practical systems.