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.

21 pages, 17 figures; typos corrected and references added