This thesis describes a continuous extrusion process for manufacturing low-density, microcellular polystyrene foam sheets using carbon dioxide as a blowing agent. Microcellular polymer foams are characterized by a cell density greater than 10⁹ cells/cm³ and a cell size on the order of 10 μm. To date, most research on the continuous processing of microcellular polymer foams has focused on nucleation and cell growth phenomena. Little work has been done on their shaping aspect. Two extrusion systems; namely, a single-screw extrusion system and a tandem extrusion system with two die designs were used in this research. The systems were designed and analyzed based on an axiomatic design framework. Detailed design and construction of their components, which include an extruder screw for the second extruder, dies, a diffusion enhancing device, a heat exchanger, a cooling mandrel, and a take-up roll system, were carried out subsequently. Critical experiments were conducted to evaluate the performance of the systems. PS foam sheets with a cell density in the range of 10$\sp9$ to 10¹⁰ cells/cm³, a controlled volume expansion ratio in the range of 2.1 to 17.9, and a uniform sheet thickness were successfully obtained from the designed tandem foam extrusion system.