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Run-I results from the CMS collaboration show an excess of events in thedecay $h \to \mu \tau_e$ with a local significances of 2.4$\sigma$. This couldbe the first hint of flavour violation in the Higgs sector. We summarise thebounds on the flavour violating Yukawa couplings from direct searches, lowenergy measurements and projected future experiments. We discuss thesensitivity of upcoming HL-LHC runs and future lepton colliders in measuringlepton-flavour violating couplings using an effective field theory framework.For the HL-LHC we find limits on BR($h\to \mu \tau$) and BR$(h\to e \tau)\lesssim \mathcal{O}(0.5)\%$ and on BR($h \to e \mu) \lesssim\mathcal{O}(0.02)\%$. For an ILC with center-of-mass energy of 1 TeV we expectBR($h \to e \tau$) and BR($h\to \mu \tau$) to be measurable down to$\mathcal{O}(0.2)\%$.

The Lepton Flavour Violating Higgs Decays at the HL-LHC and the ILC