It is shown how the Attainable Region method may be used to synthesize both chemical reactors and more general systems. This is done by using geometric ideas to first define the allowable processes as vectors in a space of the basic variables of the problem. A set of results is used to construct a region which satisfies the necessary conditions for the Attainable Region, that is the set of all possible outcomes using the allowable processes in a steady state system. The properties of the boundary of this region are of particular importance. In particular, the boundary is the union of surfaces on which single processes occur. Smooth intersections between single process surfaces may represent curves along which two or more processes occur simultaneously. Paths are traced along which one can move from a feed point(s) to other points on the boundary of the region; the paths can then be interpreted in terms of a structure, which we call the optimal structure, which will allow us to achieve any point on the boundary of the region. As these optimal structures arise from the solution of the problem they have actually been synthesized. It is further shown how these ideas might be used even for situations where the dimensionality of the problem makes it unlikely that one could find the full Attainable Region.