We develop a general, non-probabilistic model of prediction which is suitable for assessing the (un)predictability of individual physical events. We use this model to provide, for the first time, a rigorous proof of the unpredictability of a class of individual quantum measurement outcomes, a well-known quantum attribute postulated or claimed for a long time. We prove that quantum indeterminism - formally modelled as value indefiniteness - is incompatible with the supposition of predictability: measurements of value indefinite observables are unpredictable. The proof makes essential use of a strengthened form of the Kochen-Specker theorem proven previously to identify value indefinite observables. This form of quantum unpredictability, like the Kochen-Specker theorem, relies on three assumptions: compatibility with quantum mechanical predictions, non-contextuality, and the value definiteness of observables corresponding to the preparation basis of a quantum state. We explore the relation between unpredictability and incomputability and show that the unpredictability of individual measurements of a value indefinite quantum observable complements, and is independent of, the global strong incomputability of any sequence of outcomes of this particular quantum experiment. Finally, we discuss a real model of hypercomputation whose computational power has yet to be determined, as well as further open problems. ; Comment: 19 pages