Scattering amplitudes play a central role in our understanding of the fundamental laws of nature. On the one hand they constitute the bridge between theory and experiments, allowing to compute predictions starting from a theoretical model. On the other, they shed light on the underlying properties of quantum field theories. To both ends, analytic results for multi-particle multi-loop amplitudes are in high demand. In particular, the foremost and most urgent challenge is currently represented by five-particle amplitudes at two-loop order. Despite the dramatic recent progress, such computations remain formidable tasks. I review the cutting edge mathematical methods used in the analytic calculation of scattering amplitudes. As an application, I present the recent computation of the gluon-initiated contribution to di-photon plus jet production up to two loops in QCD.