  We evaluate the lepton flavour violating (LFV) decays of HIggs bosons
H->li+lj in several extensions of the SM, including both the effective
lagrangian approach and several specific models. For the effective lagrangian
case, we focus on dimension-six operators that induce LFV vertices for the
Higgs and or Z bosons. For those operators whose coefficients can not be
constrained by present data, LFV Higgs decays can have a B.R. of order
0.1-0.01. Those cases when current bounds apply, induce strong bounds for e-mu
transitions, and the b.r. for H->e mu is of order 10^-9; however H->tau mu, tau
e, can have B.R. of order 0.1. In the case of the two-higgs doublet model III,
we obtain a similar B.R. for H-> tau mu, tau e, whereas in the SM with massive
neutrinos and the minimal SUSY-SM the corresponding B.R. are very suppressed.
