This review focuses on the description of the main processes and materials used for the formulation of rigid polymer foams. Polyurethanes and their derivatives, as well as phenolic systems, are described, and their main components, foaming routes, end of life, and recycling are considered. Due to environmental concerns and the need to find bio-based alternatives for these products, special attention is given to a recent class of polymeric foams: tannin-based foams. In addition to their formulation and foaming procedures, their main structural, thermal, mechanical, and fire resistance properties are described in detail, with emphasis on their advanced applications and recycling routes. These systems have been shown to possess very interesting properties that allow them to be considered as potential substitutes for non-renewable rigid polymeric cellular foams.