Gravity, Twistors and Amplitudes

n-particle scattering amplitudes have been discovered that exhibit remarkable new mathematical structures. On the conceptual side, a new proof of GR and Yang-Mills uniqueness has been obtained, in which the basic cubic interaction is fixed (on-shell) from simple scaling considerations, and all other amplitudes are built from the basic cubic ones by the use of recursion relations. Thus, from this perspective, gravity and Yang-Mills theories are the simplest nontrivial quantum field theories. Furthermore, there are remarkable relations between gravity and the `square' of Yang-Mills arising from colour-kinematic duality. Unfortunately, at present we only see manifestations of this simplicity in the perturbative structure of the field equations. An open problem is to understand these structures nonperturbatively, arising from hidden structures also in fully non-linear gravity and Yang-Mills theories.