Summary: A spinless quantum covariant field \(\phi\) on Minkowski spacetime \({\mathcal{M}^{d + 1}}\) obeys the relation \(U(a, {\Lambda})\phi(x)U(a, {\Lambda})^{-1} = \phi({\Lambda}x + a)\) where \((a, {\Lambda})\) is an element of the Poincaré group \(\mathscr{P}\) and \(U : (a, {\Lambda}) \to U(a, {\Lambda})\) is an unitary representation on quantum vector states. It expresses the fact that Poincaré transformations are being unitarily implemented. It has a classical analogy where field covariance shows that Poincaré transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the \(\ast\)-operation are in conflict so that there are no \(\ast\)-covariant Voros fields, a result we found earlier. The notion of Drinfel’d twist underlying much of the preceding discussion is extended to discrete Abelian and non-Abelian groups such as the mapping class groups of topological geons. For twists involving non-Abelian groups the emergent spacetimes are nonassociative.