  Water waves are well-known to be dispersive at the linearization level.
Considering the fully nonlinear systems, we prove for reasonably smooth
solutions the optimal Strichartz estimates for pure gravity waves and the
semi-classical Strichartz estimates for gravity-capillary waves; for both 2D
and 3D waves. Here, by optimal we mean the gains of regularity (over the
Sobolev embedding from Sobolev spaces to H\"older spaces) obtained for the
linearized systems. Our proofs combine the paradifferential reductions of
Alazard-Burq-Zuily with a dispersive estimate using a localized wave package
type parametrix of Koch-Tataru.
