Abstract: The anomalous magnetic moment of the muon ($a_\mu$) has been, for two decades, one of the main probes for physics beyond the Standard Model of Elementary Particles. Recently, the theory initiative published the update of the high precision computation of this observable within the Standard Model, and soon FNAL will announce their final results (one day before this seminar!!!). The historic tension between experiment and theory has required us to improve our understanding of the fundamental forces (QED, EW, and QCD) and the methods to compute observables. Precision is the key.

The non-perturbative nature of QCD below 2 GeV has become an urgent issue to be addressed, as it is the main source of error. The hadronic section is split into two contributions: the hadronic vacuum polarization (HVP) and the hadronic light-by-light(HLbL). Our work is focused on the study and improvement of 3 pieces of HLbL: the pseudoscalar poles, the proton-box, and the tensor poles. The theoretical foundations to account for these contributions and the specific methods for computing the form factors required for each case will be discussed. Finally, an executive summary of the g - 2 theory initiative update will be mentioned.