  The existence of Beyond Standard Model (BSM) physics is firmly suggested by
both experimental observations (Dark Matter, neutrino masses) and theoretical
arguments. In the hypothesis that the scale of new physics is considerably
higher than the energies probed at colliders, we can parametrise modified
interactions induced by BSM effects among SM particles in a model-independent
framework, the Standard Model Effective Field Theory (SMEFT). Searches for
indirect evidence of new physics are conceptually different from the direct
ones that have characterised the first part of the LHC program, and both
experimental and phenomenological studies are needed in order to maximise the
chances of uncovering a BSM signal. In this thesis, several phenomenological
aspects of the SMEFT are discussed, both at present and future colliders. A
characteristic feature of modified interactions is that they can induce
unitarity violating effects which can be exploited to gain sensitivity. In this
direction, a thorough study of the top quark electroweak sector will be
presented, focusing on 2 to 2 scatterings and their embeddings in physical
processes at colliders. This analysis allows us to identify several final
states that have a good potential to explore the SMEFT parameter space and that
could be particularly relevant in a global analysis. One of the key features of
the SMEFT is indeed that deviations from the SM interactions are correlated and
global interpretations are therefore of fundamental importance. A combined
interpretation of the Higgs, top and diboson data from the LHC is here
presented and the interplay between the various datasets discussed. Finally,
the physics potential of a futuristic muon collider will be analysed, focusing
in particular on the prospects to determine the Higgs self-interactions, a task
that is arduous even in proposed 100 TeV proton colliders.
