  The study of higher-dimensional black holes is a subject which has recently
attracted a vast interest. Perhaps one of the most surprising discoveries is a
realization that the properties of higher-dimensional black holes with the
spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very
similar to the properties of the well known four-dimensional Kerr metric. This
remarkable result stems from the existence of a single object called the
principal tensor. In our review we discuss explicit and hidden symmetries of
higher-dimensional black holes. We start with the overview of the Liouville
theory of completely integrable systems and introduce Killing and Killing-Yano
objects representing explicit and hidden symmetries. We demonstrate that the
principal tensor can be used as a `seed object' which generates all these
symmetries. It determines the form of the black hole geometry, as well as
guarantees its remarkable properties, such as special algebraic type of the
spacetime, complete integrability of geodesic motion, and separability of the
Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a
discussion of different applications of the developed formalism and its
possible generalizations.
