The Standard Model (SM) serves as one the best descriptions of fundamental physics we have and the quest for its falsification has led to it being tested to an unprecedented degree. Despite its flawless performance, there are many theoretical and phenomenological indications that the SM cannot be a complete description of nature; though, so far, no direct evidence for new physics at the TeV scale has been gathered at colliders. Far from being discouraging, the precision level reached by current experiments gives us the unique opportunity to investigate the effects of new particles whose masses are far above the TeV scale, but still produce observable effects at the scales within the direct kinematical reach of the Large Hadron Collider (LHC). Unlike for direct searches, which are limited by the energy reach of the collider, indirect searches are limited only by the theoretical and experimental control over the processes under inspection. A robust understanding of Quantum Chromodynamics (QCD) is crucial in order to achieve precision theoretical predictions in the era of initial state hadron colliders such as the LHC. An important ingredient therein are the parton distribution functions (PDFs) which parameterize the proton structure in terms of its elementary quark and gluon constituents. These quantities are non-perturbative and obtained from data using a global QCD analysis. In tandem, Effective Field Theories (EFTs), provide a convenient framework to capture the indirect effects of possible BSM resonances in low energy observables. Constraints on the EFT then translate to constraints on the nature of BSM physics. This manuscript serves to marry these two endeavours. We present machine learning-based approaches to PDF determination and specifically highlight how deep learning algorithms form ideal candidates to parameterize the PDFs in an unbiased fashion. We present the NNPDF4.0 PDF set which serves as the latest and most precise determination of proton structure delivered by such a methodology. We show how a precise determination of the PDFs has important consequences on LHC phenomenology by presenting a precision determination of the strange content of the proton and a number of key phenomenological applications. We then discuss the interplay between EFT dynamics and the PDFs; analysing the extent to which the fit of PDFs may absorb possible BSM signals and assess the implications a consistent treatment of PDFs in EFT fits has on phenomenological studies. For this, we use legacy deep inelastic scattering data from HERA and later some more modern measurements from high-mass Drell-Yan observables at the Large Hadron Collider (LHC) to investigate the back-reaction of EFT dynamics on the PDFs. The considerations presented in the above study then act as an impetus to develop a methodology that is capable of simultaneously determining proton structure alongside BSM dynamics in a consistent framework. We present a novel methodology, SIMUnet, which delivers a robust and accurate determination of PDFs and general theory parameters, of which BSM dynamics are a subset. We show how this state-of-the-art methodology can, for the first time, extract and disentangle the PDFs from BSM dynamics from a global dataset paving the way for a truly global and simultaneous interpretation of indirect searches in the context of precision physics.