I will talk about the virtues of a powerful structure in gluon scattering amplitudes in Yang-Mills theories that intimately relates all contributions (planar and non-planar) to a small number of contributing graph topologies. This establishes a set of preferred representations for multiloop amplitudes at the integrand level. When organized in this way gluon amplitudes trivially generate gravity amplitudes in related theories. I will discuss how the existence of such a representation facilitates Yang-Mills calculation and provide pedagogic examples at tree-level, and multiloop in the maximally supersymmetric theory. I will outline some of the many mysteries that remain, and discuss promising avenues for future investigation.