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We discuss recursive algorithms for state estimation and event inference, both of which are key tasks for monitoring and control of discrete event systems. In particular, we discuss algorithms for current-, initial-, and delayed-state estimation. We also discuss implications to various pertinent properties of interest, such as detectability (i.e., the ability to determine the exact system state after a finite number of events), diagnosability (i.e., the ability to detect within finite time the occurrence/type of a fault), and opacity (i.e., the guarantee that outsiders will never be able to infer that the system state lies within a set of certain secret/critical states). The talk also briefly discusses the extension of state estimation and event inference methodologies in emerging decentralized/distributed observation settings.