We consider asynchronous diagnosis in (safe) Petri net models of distributed systems, using the partial order semantics of occurrence net unfoldings. Unlike the classical case, observability and diagnosability will appear in two different forms each: a strong form associated to interleaving semantics, and a weak form characteristic of nonsequential processes, and requiring an asynchronous progress assumption on those processes. We give algebraic characterizations for both types, and give verification methods. Sufficient conditions for strong diagnosability are derived from linear semiflows. The study of weak diagnosability leads us to the analysis of a relation in occurrence nets, first presented in \cite{CDC07}: given the occurrence of some event $a$ that \emph{reveals} $b$, the occurrence of $b$ is inevitable; here $b$ may be concurrent to, or even in the future of $a$. We show that the \emph{reveals-}relation can be effectively computed on a suitable bounded prefix of the unfolding, and show its use in asynchronous diagnosis. Based on this relation, a decomposition of the Petri net unfolding into \emph{facets} is defined, yielding an abstraction technique that preserves and reflects maximal partially ordered runs.