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We describe a computational framework for computing hybridÌýtraveling-standing waves that return to a spatial translation of theirÌýinitial conditions at a later time. We introduce two parameters toÌýdescribe these waves, and explore bifurcations from pure traveling orÌýpure standing waves to these more general solutions of theÌýfree-surface Euler equations.ÌýÌýNext, we combine Floquet theory in time and Bloch theory in space toÌýstudy the stability of traveling-standing waves to harmonic andÌýsubharmonic perturbations. For the latter, we have developed newÌýboundary integral methods for the spatially quasi-periodicÌýDirichlet-Neumann operator. While much is known about the spectralÌýstability of pure traveling waves, this is the first study of generalÌýsubharmonic perturbations of pure standing waves.Ìý Our unified approachÌýfor traveling-standing waves simplifies the eigenvalue problem thatÌýarises in the pure traveling case as well.ÌýÌýÌýWe conclude with a discussion of general quasi-periodic solutions ofÌýthe free-surface Euler equations and present preliminary calculationsÌýof some simple cases.