Since then, there has been much work searching for higher dimensional extensions of this result, in particular due to possible connections to high-temperature superconductivity. The clearest statement of the basic physical reasons to expect such an extension are due to Misguich et. al, who argued that any such system would either have long-range spin order, and hence have gapless spin wave excitations, or else would have a class of topological excitations with vanishing gap. Thus, showing this result in higher dimensions would connect directly to recent ideas on topological order in quantum systems. I will sketch my proof of this result, emphasizing connections to these basic physical ideas, and discuss more recent results on the stability of topological order. In the process, I will derive various results about locality of correlation functions in these systems.