  Despite a large experimental effort, so far no evidence for flavour-violating
decays of charged leptons such as $l_i\to l_j\gamma$ and $l_i\to l_j l_k l_k$
has been found. The absence of a signal puts very severe constraints on many
extensions of the Standard Model. Here we apply a model independent approach by
studying such decays in the Standard Model effective field theory. Going beyond
leading order in the Standard Model couplings and considering all dimension 6
operators that might lead to lepton-flavour violation, we are able to extract
limits on a large number of Wilson coefficients of such operators. We are also
able to compare the impact of particular searches and find, for example, that
flavour-violating decays of the $Z$-boson $Z\to \mu e$ are much more
constrained from low-energy experiments $\mu\to e \gamma$ than from the limits
of current and future direct searches at high energy.
