This work deals with small-amplitude three-dimensional doubly periodic travelling gravity waves on the free surface of a perfect fluid in the absence of surface tension. The fluid layer is supposed to be infinitely deep, and the flow is irrotational and subject to gravity. These waves result from the nonlinear interaction of two simply periodic travelling waves making an angle 2\(\theta\) between them. The bifurcation parameter is the horizontal phase velocity. First, the authors present a history of this problem. Then they prove the main result: the existence of nonlinear diamond waves satisfying an operator equation. It is shown that for almost all angles \(\theta\), the three-dimensional travellig waves bifurcate for a set of “good” values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane.