In this talk, I describe some of the incredible advances in our understanding of perturbative quantum field theory (QFT) and our ability to use QFT to make predictions for experiments. I will argue that the traditional tools of quantum field theory (Feynman diagrams, Lagrangians, (even fields!)) deeply obfuscate our understanding of theories involving particles with spin≥1, and motivate a reformulation of such theories without any reference to off-shell particles, gauge redundancies, ghosts, etc. Originally developed for pragmatic reasons (to make concrete predictions for collider experiments), this new formalism exposed incredible, unanticipated structure in these theories, and has led to dramatic advances in our ability to determine the S-matrix, where perturbative predictions are still represented by integrals over internal, undetermined loop momenta. Finally, I will discuss some of the recent progress in our understanding of the class of functions resulting from loop integration, and describe recent (and ongoing) progress made to improve the still-difficult problem of loop integration.