The LHC allows us to scrutinise the structure of the Standard Model of particle physics at an unprecedented level of precision. On the theoretical side, this requires very sophisticated phenomenological predictions. Their development poses very interesting theoretical challenges, which link together quantum field theory, collider phenomenology and mathematics. In this talk, I will highlight some of the recent advances that allowed to overcome limitations of standard techniques and led top a new generation of high precision predictions for the LHC. I will also illustrate how these results can be used to constrain the structure of the Standard Model, and emphasise the increasingly important role that they are playing in searches for new physics.