Polymer foams are an important class of engineering material that are finding diverse applications, including as structural parts in automotive industry, insulation in construction, core materials for sandwich composites, and cushioning in mattresses. The vast majority of these manufactured foams are homogeneous with respect to porosity and structural properties. In contrast, while cellular materials are also ubiquitous in nature, nature mostly fabricates heterogeneous foams, e.g., cellulosic plant stems like bamboo, or a human femur bone. Foams with such engineered porosity distribution (graded density structure) have useful property gradients and are referred to as functionally graded foams. Functionally graded polymer foams are one of the key emerging innovations in polymer foam technology. They allow enhancement in properties such as energy absorption, more efficient use of material, and better design for specific applications, such as helmets and tissue restorative scaffolds. Here, following an overview of key processing parameters for polymer foams, we explore recent developments in processing functionally graded polymer foams and their emerging structures and properties. Processes can be as simple as utilizing different surface materials from which the foam forms, to as complex as using microfluidics. We also highlight principal challenges that need addressing in future research, the key one being development of viable generic processes that allow (complete) control and tailoring of porosity distribution on an application-by-application basis.