  We review how the muon anomalous magnetic moment ($g-2$) and the quest for
lepton flavor violation are intimately correlated. Indeed the decay $\mu \to e
\gamma$ is induced by the same amplitude for different choices of in- and
outgoing leptons. In this work, we try to address some intriguing questions
such as: Which hierarchy in the charged lepton sector one should have in order
to reconcile possible signals coming simultaneously from $g-2$ and lepton
flavor violation? What can we learn if the $g-2$ anomaly is confirmed by the
upcoming flagship experiments at FERMILAB and J-PARC, and no signal is seen in
the decay $\mu \rightarrow e\gamma$ in the foreseeable future? On the other
hand, if the $\mu \rightarrow e\gamma$ decay is seen in the upcoming years, do
we need to necessarily observe a signal also in $g-2$?. In this attempt, we
generally study the correlation between these observables in a detailed
analysis of simplified models. We derive master integrals and fully analytical
and exact expressions for both phenomena, and adress other flavor violating
signals. We investigate under which conditions the observations can be made
compatible and discuss their implications. Lastly, we discuss in this context
several extensions of the SM, such as the Minimal Supersymmetric Standard
Model, Left-Right symmetric model, $B-L$ model, scotogenic model, two Higgs
doublet model, Zee-Babu model, 331 model, and $L_{\mu}-L_{\tau}$, dark photon,
seesaw models type~I, II and III, and also address the interplay with $\mu \to
eee$ decay and $\mu-e$ conversion.
