  We systematically discuss the consequences of genuine dimension-six Higgs
operators. These operators are not subject to stringent constraints from
electroweak precision data. However, they can modify the couplings of the Higgs
boson to electroweak gauge bosons and, in particular, the Higgs
self-interactions. We study the sensitivity to which those couplings can be
probed at future $\ee$ linear colliders in the sub-TeV and in the multi-TeV
range. We find that for $\sqrt s=500$ GeV with a luminosity of 1 ab$^{-1}$ the
anomalous $WWH$ and $ZZH$ couplings may be probed to about the 0.01 level, and
the anomalous $HHH$ coupling to about the 0.1 level.
