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When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes are critical solutions to Einstein’s equations of general relativity. They exhibit interesting phenomena that are not present in more slowly rotating black holes. I will introduce recent work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss the connection with previously known axisymmetric instabilities as well as with late-time power law tails in gravitational radiation.