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.
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