Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general. Joint work with Timothy Buttsworth.