  Motivated by the measurements of $b\to s\ell^+\ell^-$ transitions, including
$R(K)$ and $R(K^*)$, we examine lepton flavour (universality) violation in $B$
decays and its connections to $\mu\to e\gamma$ in generic leptoquark models.
Considering all 10 representations of scalar and vector leptoquarks under the
Standard Model gauge group we compute the tree-level matching for semileptonic
$b$-quark operators as well as their loop effects in
$\ell\to\ell^\prime\gamma$. In our phenomenological analysis we correlate
$R(K)$, $R(K^*)$ and the other $b\to s\mu^+\mu^-$ data to $\mu\to e\gamma$ and
$b\to s\mu e$ transitions for the three leptoquark representations that
generate left-handed currents in $b\to s\ell^+\ell^-$ transitions and,
therefore, provide a good fit to data. We find that while new physics
contributions to muons are required by the global fit, also couplings to
electrons can be sizeable without violating the stringent bounds from $\mu\to
e\gamma$. In fact, if the effect in electrons in $b\to s\ell^+\ell^-$ has
opposite sign than the effect in muons the bound from $\mu\to e\gamma$ can
always be avoided. However, unavoidable effects in $b\to s\mu e$ transitions
(i.e. $B_s\to\mu e$, $B\to K\mu e$, etc.) appear which are within the reach of
LHCb and BELLE II.
