We set up a hierarchy of Petri net languages based on the properties of accepting subsets. With each more or less classical subclass of subsets of N^m --- finite, ideal (or upper), semi-cylindrical, star-free, recognizable, rational (or semilinear) subsets --- we associate the class of Petri net languages whose set of accepting states belongs to the class. We compare the related Petri net languages. For arbitrary or $\lambda$-free Petri net languages, the above hierarchy collapses: one does not increase the generality by considering semilinear accepting sets instead of the usual finite ones. However, for free-labeled and deterministic Petri net languages, we show that one gets new distinct subclasses of languages, for which several decidability problems become solvable.