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Model-based diagnosis has always been conceived as set-oriented, meaning that a candidate is a set of faults, or faulty components, that explains a collection of observations. This perspective applies equally to both static and dynamical systems. Diagnosis of discrete-event systems (DESs) is no exception: a candidate is traditionally a set of faults, or faulty events, occurring in a trajectory of the DES that conforms with a given sequence of observations. As such, a candidate does not embed any temporal relationship among faults, nor does it account for multiple occurrences of the same fault. To improve diagnostic explanation and support decision making, a sequence-oriented perspective to diagnosis of DESs is presented, where a candidate is a sequence of faults occurring in a trajectory of the DES, called a fault sequence. Since a fault sequence is possibly unbounded, as the same fault may occur an unlimited number of times in the trajectory, the set of (output) candidates may be unbounded also, which contrasts with set-oriented diagnosis, where the set of candidates is bounded by the powerset of the domain of faults. Still, a possibly unbounded set of fault sequences is shown to be a regular language, which can be defined by a regular expression over the domain of faults, a property that makes sequence-oriented diagnosis feasible in practice. The task of monitoring-based diagnosis is considered, where a new candidate set is generated at the occurrence of each observation. The approach is based on three different techniques: .1/ blind diagnosis, with no compiled knowledge, .2/ greedy diagnosis, with total knowledge compilation, and .3/ lazy diagnosis, with partial knowledge compilation. By knowledge we mean a data structure slightly similar to a classical DES diagnoser, which can be generated (compiled) either entirely offline (greedy diagnosis) or incrementally online (lazy diagnosis). Experimental evidence suggests that, among these techniques, only lazy diagnosis may be viable in non-trivial application domains.