The possible existence, form and maximum height of strictly periodic finite stationary waves on the surface of a perfect liquid are discussed. A method of successive approximation to the solution of the hydrodynamical equations is formulated, and the solution is carried to the fifth order for the case of two-dimensional waves on a deep liquid. The convergence of the method has not been established, so that the existence of truly periodic stationary waves is not beyond doubt, but the calculations provide strong presumptive evidence for their existence, and for the existence of a finite stable wave of greatest height. The crest of this wave has a right-angled nodal form, in contrast with that of the greatest stable travelling wave for which the nodal angle is 120 degrees. The maximum crest height is 0$\cdot $141 $\lambda $, where $\lambda $ is the wave-length, and the maximum trough depth is 0$\cdot $078 $\lambda $. This means that the greatest stationary waves are greater than the maximum travelling waves, the ratio being 1$\cdot $53. The motions of individual particles are studied and it is shown that particles in the surface, particularly those near the anti-nodes have large horizontal motions. For a given wave-length, the period increases with wave height. The wave pressure on a breakwater is examined, and the modification of the calculations to allow for the finite depth of water is considered. Doubly modulated oscillations in a deep rectangular tank are also briefly discussed.