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The classical Penrose inequality gives a variational characterization of Schwarzschild data as that with the minimal mass, amongst all asymptotically flat initial data sets with non- negative scalar curvature and fixed horizon area. A Penrose inequality with charge has also been established, which gives a similar variational characterization of the Reissner-Nordstrom black hole. It has been much more difficult to include angular momentum, and there have been very few results in this direction. Here we present a proof of a Penrose inequality with angu- lar momentum (also including charge), which yields a variational characterization of the Kerr (and Kerr-Newman) data. These techniques are then extended to higher dimensions to obtain Penrose-type inequalities associated with the Myers-Perry black hole (higher dimensional version of Kerr) as well as the Black Ring solution of Emparan and Reall.