  It is well known that measurements performed on spatially separated entangled
quantum systems can give rise to correlations that are non-local, in the sense
that a Bell inequality is violated. They cannot, however, be used for
super-luminal signalling. It is also known that it is possible to write down
sets of ``super-quantum'' correlations that are more non-local than is allowed
by quantum mechanics, yet are still non-signalling. Viewed as an information
theoretic resource, super-quantum correlations are very powerful at reducing
the amount of communication needed for distributed computational tasks. An
intriguing question is why quantum mechanics does not allow these more powerful
correlations. We aim to shed light on the range of quantum possibilities by
placing them within a wider context. With this in mind, we investigate the set
of correlations that are constrained only by the no-signalling principle. These
correlations form a polytope, which contains the quantum correlations as a
(proper) subset. We determine the vertices of the no-signalling polytope in the
case that two observers each choose from two possible measurements with d
outcomes. We then consider how interconversions between different sorts of
correlations may be achieved. Finally, we consider some multipartite examples.
