  These proceedings review the differential decay rates and the branching
ratios of the tau and muon decays $\tau \to \ell \ell' \ell' \nu \bar\nu$ (with
$\ell,\ell'=\mu,e$) and $\mu \to e e e \nu \bar \nu$ in the Standard Model at
NLO. These five-body leptonic decays are a tool to study the Lorentz structure
of weak interactions and to test lepton flavour universality. They are also a
source of SM background to searches for the lepton-flavour-violating decays
$\mu \to e e e$ and $\tau \to \ell \ell' \ell'$. Even if the shift in the
branching ratios induced by radiative corrections turns out to be small and of
order 1% --- mainly due to a running effect of the fine structure constant ---
locally in the phase space these corrections can reach the 5 - 10% level,
depending on the applied cuts. We found for instance that in the phase space
region where the neutrino energies are small, and the momenta of the three
charged leptons have a similar signature as in $\mu \to eee$ and $\tau \to \ell
\ell'\ell'$, the NLO corrections decrease the leading-order prediction by about
10 - 20%.
