The dual-shaped reflector synthesis problem was first solved by Kinber and Galindo in the early 1960's for the circularly symmetric-shaped reflectors. Given an arbitrary feed pattern, it was shown that the surfaces required to transform this feed pattern by geometrical optics into any specified phase and amplitude pattern in the specified output aperture are found by the integration of two simultaneous nonlinear ordinary differential equations. For the offset noncoaxial geometry, however, it is shown that the equations found by this method are partial differential equations which, in general, do not form a total differential. Hence the exact solution to this problem is believed to be generally not possible. It is also shown, however, that for many important problems the partial differential equations form a nearly total differential. It thus becomes possible to generate a smooth subreflector by integration of the differential equations and then synthesize a main reflector which gives an exact solution for the specified aperture phase distribution. The resultant energy (or amplitude) distribution in the output aperture as well as the output aperture periphery are then approximately the specified values. A representative group of important solutions are presented which illustrate the very good quality that frequently results by this synthesis method. This includes high gain, low sidelobe, near-field Cassegrain, and different (f/D) ratio reflector systems. While the above approach generates the solutions for the surfaces by numerically solving a line integral along the surface, more recent approaches develop the entire area from either an 'initial' line (usually of symmetry) or a point. The latter development is particularly useful in addressing the question of the existence of solutions - a currently controversial and very significant question. In this regard it is most important to pay careful attention to the required constraints posed in the problem.