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Abstract: We give the general Kerr–de Sitter metric in arbitrary space–time dimension , with the maximal number of independent rotation parameters. We obtain the metric in Kerr–Schild form, where it is written as the sum of a de Sitter metric plus the square of a null-geodesic vector, and in generalised Boyer–Lindquist coordinates. The Kerr–Schild form is simpler for verifying that the Einstein equations are satisfied, and we have explicitly checked our results for all dimensions . We discuss the global structure of the metrics, and obtain formulae for the surface gravities and areas of the event horizons. We also obtain the Euclidean-signature solutions, and we construct complete non-singular compact Einstein spaces on associated bundles over , infinitely many for each odd . [Copyright &y& Elsevier]