  While the muon $g-2$ anomaly can be successfully explained by some new
physics models, most of them are severely constrained by the $\mu \to e \gamma$
bound. This tension is more transparent from the effective field theory
perspective, in which the two phenomena are encoded in two very similar
operators. However, with the ${\cal O}(1)$ Wilson coefficients, the current
upper bound on $\mu \to e \gamma$ indicates a new-physics cutoff scale five
orders smaller than that needed to eliminate the $(g-2)_\mu$ anomaly. By
summarizing all the formulae from the one-loop contributions to the muon $g-2$
with the internal-particle spin not larger than 1, we point out two general
methods to reconcile the conflict between the muon $g-2$ and $\mu \to e
\gamma$: the GIM mechanism and the non-universal couplings. For the latter
method, we use a simple scalar leptoquark model as an illustration.
