The 4D rotating black hole described by the Kerr geometry possesses many of what was called by Chandrasekhar "miraculous" properties. Most of them can be related to the existence of a fundamental hidden symmetry called the principal conformal Killing-Yano (PCKY) tensor. In my talk I shall demonstrate that, in this context, four dimensions are not exceptional and that the (spherical horizon topology) higher-dimensional rotating black holes are very similar to their four-dimensional cousins. Namely, I shall present the most general spacetime admitting the PCKY tensor and show that is possesses the following properties: 1) it is of the algebraic type D, 2) it allows a separation of variables for the Hamilton-Jacobi, Klein-Gordon, Dirac, gravitational, and stationary string equations, 3) the geodesic motion in such a spacetime is completely integrable, 4) when the Einstein equations with the cosmological constant are imposed the metric becomes the Kerr-NUT-(A)dS spacetime. Some of these properties remain valid even when one includes the electromagnetic field.
