Observing new physics at hadron colliders requires a detailed description of multi-particle scattering events. It is not practical to evaluate Feynman diagrams directly for all significant processes. Moreover, adding all diagrams reveals many cancellations: scattering amplitudes in theories such as QCD and gravity take remarkably simple forms. This simplicity is a clue that the perturbative theory is best understood without reference to Feynman diagrams. In fact, it has recently become possible to explain some of this simplicity and derive amplitudes efficiently and elegantly. Symmetry can be preserved by working recursively on the mass shell: amplitudes are expressed in terms of other amplitudes, based on analysis of their singularities. I will describe these insights and some applications.