The authors prove that Alexander-Spanier cohomology \(H^n(X;G)\) with coefficients in a topological Abelian group \(G\) is isomorphic to the group of isomorphism classes of principal bundles with certain Abelian structure groups. The result holds if either \(X\) is a CW-space and \(G\) arbitrary or if \(X\) is metrizable or compact Hausdorff and \(G\) an ANR.