  Axion-like particles (ALPs) with lepton-flavor violating couplings can be
probed in exotic muon and tau decays. The sensitivity of different experiments
depends strongly on the ALP mass and its couplings to leptons and photons. For
ALPs that can be resonantly produced, the sensitivity of three-body decays such
as $\mu\to 3e$ and $\tau\to 3\mu$ exceeds by many orders of magnitude that of
radiative decays like $\mu\to e\gamma$ and $\tau\to\mu\gamma$. Searches for
these two types of processes are therefore highly complementary. We discuss
experimental constraints on ALPs with a single dominant lepton-flavor violating
coupling. Allowing for one or more such couplings offers qualitatively new ways
to explain the anomalies related to the magnetic moments of the muon and
electron.
