Fractal solutions of the inviscid water-wave problem are presented. For gravity waves (neglecting surface tension) free surfaces with fractal dimensions 2 1 4 and 2 1 3 are obtained. For capillary waves (neglecting gravity), subfractal free surfaces with dimension 2 are shown to exist. However, the situation is reversed if one considers time series of the surface elevation taken at a fixed point. In this case the capillary wave solution produces graphs with dimension 1 1 12, whereas the graph for gravity waves has dimension 1.