The weight of evidence supporting the case for hypercomputation is compelling. We examine some 20 physical and mathematical models of computation that are either known or suspected to have super-Turing or hypercomputational capabilities, and argue that there is nothing in principle to prevent the physical implementation of hypercomputational systems. Hypercomputation may indeed be intrinsic to physics; recursion ‘emerges’ from hypercomputation in the same way that classical physics emerges from quantum theory as scale increases. Furthermore, even if hypercomputation were one day shown to be physically infeasible, there would still remain a role for hypercomputation as an organising principle for advanced research.