After the final release of results from the Fermilab Muon $g-2$ experiment, the experimental world average has reached a precision of 124 parts per billion, allowing for a Standard-Model precision test below 0.2 parts per million if the theory prediction could be evaluated at a similar level of precision as the experimental result. At present, this is not the case, diluting the sensitivity to potential beyond-the-Standard-Model degrees of freedom. Apart from improved lattice-QCD calculations, a number of ongoing efforts using data-driven methods aim at improving the current puzzling situation, to ideally arrive at two independent determinations of the critical hadronic contributions and thereby increase confidence in the resulting Standard-Model prediction. Here, we present a summary of the current status, both for hadronic vacuum polarization and hadronic light-by-light scattering, discuss the most pressing challenges, and give an overview over the various avenues for future improvements.