We prove that \(\omega\)-languages of (non-deterministic) Petri nets and \(\omega\)-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of \(\omega\)-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of \(\omega\)-languages of (non-deterministic) Turing machines which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal \(\alpha < \omega\_1^{{\rm CK}} \) there exist some \({\bf \Sigma}^0\_\alpha\)-complete and some \({\bf \Pi}^0\_\alpha\)-complete \(\omega\)-languages of Petri nets, and the supremum of the set of Borel ranks of \(\omega\)-languages of Petri nets is the ordinal \(\gamma\_2^1\), which is strictly greater than the first non-recursive ordinal \(\omega\_1^{{\rm CK}}\). We also prove that there are some \({\bf \Sigma}\_1^1\)-complete, hence non-Borel, \(\omega\)-languages of Petri nets, and that it is consistent with ZFC that there exist some \(\omega\)-languages of Petri nets which are neither Borel nor \({\bf \Sigma}\_1^1\)-complete. This answers the question of the topological complexity of \(\omega\)-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].