A new processing system for the extrusion of microcellular polymer sheets is presented. Specifically, the detailed design of a shaping and cell growth control system is discussed in the context of an overall extrusion system design with particular emphasis on the system level functional requirements of cell nucleation, cell growth, and shaping. The principle of the basic extrusion system design is to shape a nucleated polymer/gas solution flow under pressure and accurate temperature control. In this way, the initial cell growth is controlled so as to prevent degradation of the nucleated cell density during shaping. Two foaming die designs for satisfying the initial shaping and cell growth requirements are presented. Critical experiments are then presented which verified the concept of shaping a nucleated polymer/gas solution. Moreover, these experiments demonstrated the feasibility of the overall microcellular polymer sheet extrusion system design.