The authors establish the notion of a covariant representation (\(\pi\),w) for a coaction \(\delta\) of a locally compact group G on a \(C^*\)-algebra A. This unitary W is related to a representation of the Fourier algebra A(G) [Y. Nakagami and M. Takesaki, Duality for crossed products of von Neumann algebras, Theorem A.1(b), Lect. Notes Math. 731, Berlin (1979; Zbl 0423.46051)], and \(\pi\) is a representation of A with \((\pi \otimes t)^-(\delta (a))=W(\pi (a)\otimes 1)W^*,\) a in A. Then there is a one-to-one correspondence between representations \(\pi\times W\) of the crossed coproduct \(A\times_{\delta}G\) and the covariant representation (\(\pi\),W). The authors give the definitions of pointwise unitary coaction and locally unitary coaction. If \(\delta\) is a pointwise unitary coaction on the CCR-algebra A with Hausdorff spectrum \(A^{{\hat{\;}}}\), then the map p: \(\pi\times W\to \pi\) is a continuous surjection of \(A\times_{\delta}G^{{\hat{\;}}}\quad to\quad A^{{\hat{\;}}}\) and \(p: A\times_{\delta}G^{{\hat{\;}}}\to A^{{\hat{\;}}}\) is a principal G-bundle. Moreover, if \(\delta\) is locally unitary, its G-bundle becomes locally trivial. Isomorphic classes of locally trivial principal G-bundles are completely invariant for exterior equivalence classes of locally unitary coactions on A. Although the results are analogous in the case of actions, the proofs are quite different and more difficult. These results are applied by E. C. Gootman and A. J. Lazar [“Applications of noncommutative duality to crossed product algebras determined by an action or coaction”, Proc. Lond. Math. Soc., III. Ser. to appear (Zbl 0651.46065)] to the proof that the crossed product \(A\times_{\alpha}G\) is an AF-algebra if G is a compact group and A is an AF-algebra of type I.