This project focuses on developing quantum field theory methods and applying them to the phenomenology of elementary particles. At the Large Hadron Collider (LHC) our current best theoretical understanding of particle physics is being tested against experiment by measuring e.g. properties of the recently discovered Higgs boson. With run two of the LHC, currently underway, the experimental accuracy will further increase. Theoretical predictions matching the latter are urgently needed. Obtaining these requires extremely difficult calculations of scattering amplitudes and cross sections in quantum field theory, including calculations to correctly describe large contributions due to long-distance physics in the latter. Major obstacles in such computations are the large number of Feynman diagrams that are difficult to handle, even with the help of modern computers, and the computation of Feynman loop integrals. To address these issues, we will develop innovative methods that are inspired by new structures found in supersymmetric field theories. We will extend the scope of the differential equations method for computing Feynman integrals, and apply it to scattering processes that are needed for phenomenology, but too complicated to analyze using current methods. Our results will help measure fundamental parameters of Nature, such as, for example, couplings of the Higgs boson, with unprecedented precision. Moreover, by accurately predicting backgrounds from known physics, our results will also be invaluable for searches of new particles.