  This Report summarizes some of the activities of the HiggsTools Initial
Training Network working group in the period 2015-2017. The main goal of this
working group was to produce a document discussing various aspects of
state-of-the-art Higgs physics at the Large Hadron Collider (LHC) in a
pedagogic manner.
  The first part of the Report is devoted to a description of phenomenological
searches for New Physics at the LHC. As the experimental measurements become
more and more precise, there is a pressing need for a consistent framework in
which deviations from the SM predictions can be computed precisely. We
critically review the use of the \k{appa}-framework, fiducial and simplified
template cross sections, effective field theories, pseudo-observables and
phenomenological Lagrangians.
  In the second part of the Report, we propose $\varphi_{\eta}^*$ as a new and
complementary observable for studying Higgs boson production at large
transverse momentum in the case where the Higgs boson decays to two photons. We
make a detailed study of the phenomenology of the $\varphi_{\eta}^*$ variable,
contrasting the behaviour with the Higgs transverse momentum distribution using
a variety of theoretical tools including event generators and fixed order
perturbative computations.
