  We give the general Kerr-de Sitter metric in arbitrary spacetime dimension
D\ge 4, with the maximal number [(D-1)/2] 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 D\le 11. 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 S^{D-2} bundles
over S^2, infinitely many for each odd D \ge 5.
