The diffraction pattern of an aperture with amplitude distribution A(x) and phase function ψ(x) is examined for two cases: (1) controlled variation of the amplitude distribution with known uniform phase; (2) controlled variation of phase with known amplitude distribution. For Case 1, it is shown that a symmetrically tapered amplitude distribution produces a diffraction pattern with reduced secondary lobes, while an asymmetric amplitude distribution produces a shaped pattern. Thus, antenna beam shaping may be achieved by controlled variation of amplitude distribution in an aperture in which there is uniform phase. For Case 2, it is shown that a specified beam shape may be approximated by controlled variation of phase in the aperture when the amplitude distribution is known. The method of calculation is an extension of the method originally formulated by Chu for calculating cylindrical reflectors to transform a given primary radiation pattern to a specified shaped pattern. A general formulation of Chu's method for an amplitude distribution on a curved surface is developed. The theoretical results are applied to the design of progressive-phase antennas and experimental results given.