The scientific approach to understanding the laws of nature is based on the comparison between theory and experiment. In laypersons' terms, a theory is a set of rules which describe mathematically how we think things work. We use these rules to predict the outcome of a certain experiment, and the comparison against the actual results of the experiment may disprove or uphold the theory. Particle physics is concerned with the tiniest building blocks of the universe - the fundamental particles - and the way they interact. Our best description of fundamental particles is given by the Standard Model of particle physics (SM), which treats particles as oscillations of "quantum fields" permeating the space-time. The spectacular success of the SM at describing the microscopic world is one of humanity's greatest intellectual feats. Yet, this theory fails to address a variety of theoretical concerns and observed phenomena|gravity, to say the most obvious. Understanding the limitations of the SM and constraining its extensions is of primary importance. Scattering amplitudes are the bridge between theory and experiments in Quantum Field Theories (QFTs). Roughly speaking, the amplitude of a scattering process encodes its probability distribution: for a given initial state|say two colliding protons - the scattering amplitude tells us how likely the production of certain other particles is according to the theory. The rules of QFT are however complicated, and scattering amplitudes can only be computed approximately as series in the coupling constants which weigh the interactions. We know - at least in principle - how to compute each term of the series, and including more terms makes the prediction more accurate. The computation however becomes more and more difficult as the order in the couplings - also called the "loop order" - or the number of particles increase. In practice, we need as many terms as is necessary to make the theoretical uncertainty comparable with the experimental one, so that the comparison is statistically significant. Exploiting fully the physics potential of CERN's Large Hadron Collider requires predictions at the Next-to-Next-to-Leading Order (NNLO) in the coupling of the strong interactions. This goal has already been reached for many 2 -> 1 and 2 -> 2 processes. Processes with three particles in the final state are however of great interest, as they would allow for precise measurements of the strong coupling constant and of its scaling, in-depth studies of the Higgs couplings, better background estimates for yet unknown phenomena, and more. The main bottleneck towards NNLO predictions for 2 -> 3 processes is the analytic computation of two-loop five-particle scattering amplitudes. The most difficult part of computing a scattering amplitude is the computation of the Feynman integrals appearing in it. My collaborators and I computed the missing and most complicated set of massless two-loop five-particle Feynman integrals. This opened the doors to the computation of the amplitude for any process involving five massless particles at two-loop order. Such processes feature prominently in the LHC physics program. Cases in point are three-jet, three-photon, and di-photon + jet production. In order to compute these integrals we made use of cutting-edge mathematical techniques, and proposed a new strategy which has already been applied to other difficult problems. Armed with analytic expressions for the Feynman integrals, we tackled the amplitudes. The challenge is one of enormous algebraic complexity. We developed a work ow based on the recent idea of evaluating the rational functions in the intermediate expressions numerically over finite fields. The analytic expression of the final result is then reconstructed by "bootstrapping" an Ansatz or through reconstruction algorithms. Before considering the SM, we tested our approach on the amplitudes in two supersymmetric theories: N = 4 super Yang-Mills theory and N = 8 supergravity. These were the very first complete five-particle scattering amplitudes to be computed analytically at two loops. Although these models do not seek to describe physical particles and forces, they are of great interest. They give precious insights into hidden structures of QFT in general and - thanks to their simplicity and elegance - they are a perfect testing ground for new techniques and ideas which can be later applied to the SM. The successful computation of the supersymmetric amplitudes showed that our technology was mature enough to face the SM. We therefore computed the two-loop amplitude describing the scattering of five positive-helicity gluons in Quantum Chromodynamics (QCD), the part of the SM which describes the strong interactions. Despite the leap in complexity with respect to the supersymmetric theories, we managed to find an extremely compact and elegant analytic expression. Having compact results for the amplitudes is not only a theorist's delight, but is crucial for their use in phenomenology. The simplicity of the expression allowed us to notice that certain parts of the amplitude enjoy an unexpected property: they are invariant under conformal symmetry. We identified the origin of this property in the conformal invariance of the gluonic amplitudes in QCD at one loop, which we proved for any number of gluons. After the publication of the results presented in this thesis there has been a dramatic progress. Several other two-loop five-particle amplitudes have become available analytically, and this has already led to the first theoretical prediction at NNLO in QCD, for three-photon production. Partly using methods similar to those presented in this thesis, many more results are sure to follow in the near future
