Periodic traveling waves are computed on parameterized interfaces, which are not functions of the horizontal coordinate(s). These overturned traveling waves are computed on one and two-dimensional interfaces, on a classic interface between two fluids as well as on boundary formed by a hydroelastic ice sheet. Numerical continuation procedures are coupled with local and global bifurcation theorems. Extreme wave types and bifurcation surfaces are presented. The prospects for stability of overturned traveling waves are discussed.