We consider asynchronous diagnosis in (safe) Petri net models of distributed systems, using the partial order semantics of occurrence net unfoldings. Both the observability and diagnosability properties will appear in two different forms, depending on the semantics chosen: strong observability and diagnosability are the classical notions from the state machine model and correspond to interleaving semantics in Petri nets. By contrast, the weak form is linked to characteristics of nonsequential processes, and requires an asynchronous progress assumption on those processes. We give algebraic characterizations for both types, and give verification methods. The study of weak diagnosability leads us to the analysis of a relation in occurrence nets, first presented in [14]: given the occurrence of some event a that reveals b, the occurrence of b is inevitable. Then b may already have occurred, be concurrent to, or even in the future of a. We show that the reveals-relation can be effectively computed recursively-for each pair, a suitable finite prefix of bounded depth is sufficient-, and show its use in asynchronous diagnosis. Based on this relation, a decomposition of the Petri net unfolding into facets is defined, yielding an abstraction technique that preserves and reflects maximal partially ordered runs.