The Kerr geometry is that remarkable solution of Einstein's equation which is believed to universally describe rotating black holes (BHs) up in the sky. General Relativity implies that, near certain critical points, excitations of Kerr are governed by an emergent infinite-dimensional conformal symmetry. In (near-) extremal, rapidly rotating BHs it arises as an asymptotic symmetry of the near-horizon region, while in non-extremal BHs it arises in regions of phase space corresponding to critical excitations. These facts have motivated the Kerr/CFT conjecture which asserts that Kerr is holographically dual to a conformal field theory. In the talk I will describe the symmetries and some of their implications for BH theory, ranging from the information paradox to horizon instability. I will also discuss their consequences for observation, where they provide analytical predictions, and suggestions, for the recently successful gravitational-wave and optical BH observatories. I will comment on some open problems.
