[Hypercomputation][1] refers to models of computation that are not possible to simulate using Turing machines.  (Hypercomputers are not necessarily physically realisable!)  Some hypercomputers have access to a resource that allows the [Halting Problem][2] for standard Turing machines to be solved.  Call this a "superpower": a hypercomputer with a superpower can decide whether any standard Turing machine terminates.

>What kinds of "superpowers" do hypercomputers use?

[Ed Blakey's thesis][3] sets up a formal framework to classify some of the major kinds of resources used in hypercomputing, but it does not try to provide a comprehensive survey of superpowers.  I am not interested in a list of hypercomputers (there is a nice list in the Wikipedia article), but in understanding what "special sauce" each model uses, perhaps thought of as a unique kind of resource.

This question is inspired by https://cstheory.stackexchange.com/questions/5927/how-fundamental-is-undecidability-closed.  Also related is https://cstheory.stackexchange.com/questions/88/what-would-it-mean-to-disprove-church-turing-thesis which generated lots of interesting discussion, and https://cstheory.stackexchange.com/questions/3498/are-there-any-models-of-computation-currently-being-studied-with-the-possibility.

  [1]: http://en.wikipedia.org/wiki/Hypercomputation
  [2]: http://en.wikipedia.org/wiki/Halting_problem
  [3]: http://ora.ox.ac.uk/objects/uuid:5db40e2c-4a22-470d-9283-3b59b99793dc