Powerful computational tools and new structures for perturbative scattering amplitudes in QFT have been discovered in recent years. Apart from playing a central role in high-energy particle physics, these developments have important implications for formal QFT, quantum gravity, string theory and even new directions in mathematics. I will review some recent progress, focusing on a new formulation of scattering of gluons, gravitons and their connections, as well as geometries underlying QFT and strings, which may hint at a ``theory at infinity” for their S-matrices.