Summary: The asynchronous fault diagnosis problem involves detecting the occurred faults in a system in the scenario that one asynchronously initiates a diagnosis agent with the system. This research investigates the asynchronous fault diagnosis problem of a system represented by labeled Petri nets (LPNs). An important property named asynchronous diagnosability of an LPN plant is formulated and studied which indicates that the detection of a fault can be achieved within a finite number of observations in the context of asynchronous diagnosis. This paper first introduces a structure named an asynchronous basis diagnoser, based on which the asynchronous diagnosis problem of an LPN plant can be addressed. Moreover, an approach is presented which leverages the asynchronous basis diagnoser to identify the asynchronous diagnosability. If an LPN plant is not asynchronously diagnosable, a supervisor is then designed which constraints the plant’s behavior such that the controlled system satisfies the requirement of asynchronous diagnosability.