Our present knowledge of elementary particles and their interactions is described in the so-called Standard Model (SM) of particle physics. The discovery of a scalar boson with a mass of 125 GeV, corresponding to the Higgs boson in the SM, is the most remarkable confirmation of the predictions of the SM. In recent years, the direct discovery of new physics states has become less frequent, and there is no sign of new physics beyond the SM so far. Therefore, new physics states might be on a very high energy scale that current experiments cannot achieve. However, it is possible that new BSM states can manifest themselves via tree-level exchange or loop-level effects and then indirectly affect SM precision measurements at the low-energy scale. Recently, many indirect searches for the hints of new physics states have been performed and received much attention. Following this aspect, Effective Field Theories (EFT) raise as a robust theoretical framework that allows for hunting for new physics effects in a model-independent way (bottom-up approach) or connecting the UV complete model with the theories at low energies and their precision measurements (top-down approach). In this thesis, we have developed functional methods for EFT top-down approach and applied our techniques to study axion physics, QFT anomalies and some recent flavour discrepancies in the SM.The first part of this thesis is devoted to studying the path integral formulation of one-loop matching calculations. This technique provided an elegant way to match a UV model onto an EFT Lagrangian resulting from integrating out a heavy degree of freedom. Such matching computations are the most crucial point in the EFT top-down paradigm. Ultimately, matching by the functional method will generate universal structures, which makes repeated evaluation of the loop integrals redundant. This set of universal structures can be pre-computed once and for all, forming the so-called Universal One-Loop Effective Action (UOLEA). Starting from the UOLEA, one-loop matching calculations are reduced to an algebraic manipulation of matrix traces. In this part, we studied all universal structures arising from integrating out heavy fermions (Dirac or Majorana) with scalar, pseudo-scalar, vector, and axial-vector couplings. The final results provided an analytic expression for all coefficients of the universal structures. They served as a reference to cross-check if one would like to decouple heavy chiral fermions.The second part of this thesis is devoted to studying QFT anomalies and axion physics. First, we utilised the power of the functional method to re-compute the well-known QFT anomalies (e.g. chiral anomaly, Weyl anomaly, axial gravitational anomaly). Second, we built the Effective Field Theories in which a chiral fermion, charged under both gauge and global symmetries, is integrated out. The challenge of this exercise is that the derivative couplings of the Goldstone bosons to the fermions will lead to severe divergences and ambiguities when building the EFT. We then precisely solve these difficulties within the path integral formalism by explicitly evaluating the anomaly-related operators and adapting the anomalous Ward identities to the EFT context. Our results provide a systematic and consistent methodology to build EFTs involving anomalous symmetries, as required for axion searches.The third part of this thesis briefly studied flavour anomalies (e.g. B-anomalies, one-loop muon anomaly) within the EFT framework. We performed the one-loop matching of Leptoquark models onto the EFT Lagrangian. Our results cross-check previous literature calculations and provide a reference for further computations. We performed one-loop benchmarks to show that if the muon anomalies have a common origin and do not arise at the tree-level, they will be discovered or excluded in the near future experiments.