  We review the present status of the Standard Model calculation of the
anomalous magnetic moment of the muon. This is performed in a perturbative
expansion in the fine-structure constant $\alpha$ and is broken down into pure
QED, electroweak, and hadronic contributions. The pure QED contribution is by
far the largest and has been evaluated up to and including
$\mathcal{O}(\alpha^5)$ with negligible numerical uncertainty. The electroweak
contribution is suppressed by $(m_\mu/M_W)^2$ and only shows up at the level of
the seventh significant digit. It has been evaluated up to two loops and is
known to better than one percent. Hadronic contributions are the most difficult
to calculate and are responsible for almost all of the theoretical uncertainty.
The leading hadronic contribution appears at $\mathcal{O}(\alpha^2)$ and is due
to hadronic vacuum polarization, whereas at $\mathcal{O}(\alpha^3)$ the
hadronic light-by-light scattering contribution appears. Given the low
characteristic scale of this observable, these contributions have to be
calculated with nonperturbative methods, in particular, dispersion relations
and the lattice approach to QCD. The largest part of this review is dedicated
to a detailed account of recent efforts to improve the calculation of these two
contributions with either a data-driven, dispersive approach, or a
first-principle, lattice-QCD approach. The final result reads
$a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$ and is smaller than the
Brookhaven measurement by 3.7$\sigma$. The experimental uncertainty will soon
be reduced by up to a factor four by the new experiment currently running at
Fermilab, and also by the future J-PARC experiment. This and the prospects to
further reduce the theoretical uncertainty in the near future-which are also
discussed here-make this quantity one of the most promising places to look for
evidence of new physics.
