It is astonishing how the variety of Nature, from all the living species to oceans, mountains and stars, results from different rearrangements of the same few building blocks, the elementary particles. The standard model (SM) of particle physics has been incredibly effective so far in describing the interactions among our fundamental constituents, which ultimately determine significant properties of ourselves and our universe. The 2012 discovery of the Higgs boson at the LHC experiments has marked the completion of the SM picture, and the beginning of a new journey: the quest for new physics beyond the current paradigm. This quest can be pursued in many directions, including the search for deviations from the SM predictions in the interactions among the Higgs boson and the other elementary particles. In this thesis I address the interaction between the Higgs boson and the top quark, the two heaviest known elementary particles, which are at the centre of the LHC research. I promote predictions accurate at next-to-leading order (NLO) in perturbation theory for processes relevant to study this interaction at the LHC. I show how NLO accuracy is decisive in order to reduce systematic uncertainties in the theoretical computations, such as the scale dependence or the number of light quarks (flavour scheme). It is essential to control these uncertainties if we want to spot signs of new phenomena in deviations from the SM. I also discuss several observables sensitive to new physics that can be measured at the LHC. In particular, I focus on hypothetical CP-violating effects in the Higgs-top interaction, which could help to explain the imbalance between matter and antimatter observed in the universe.