Porous sandwich structure (foam) is a prerequisite in the field of construction, aerospace, and automotive sector, due to its energy-absorbing characteristics and sound-absorbing properties. The porous structure was previously difficult to manufacture, but is now feasible, thanks to additive manufacturing process or 3D printing technology. Complex shapes and structure can be produced by the novel production technique such as fused deposition modeling (FDM) known as three-dimensional printing. These studies emphasized the importance of porous structure, particularly foams, for the construction industry, automobile sector and aerospace industries for its high performance, light-weightiness, shock absorbing properties, etc. In this article, we convey the different processes as well as their main features which can be used to fabricate the foam structure having various topologies such as open cells and closed cells, with different fundamental designs and materials such as polymer, ceramic, and metals, with a particular emphasis on metallic and polymeric materials. In this regard, a comparison between the traditional foam manufacturing techniques and additively 3D printed foam structure was made, to study the various complexities associated with such method of fabrication and attempts to integrate such study have been divided into four categories: (1) architected porous structures, (2) conventional and 3D methods of polymeric and metallic foam construction, (3) foam lattice design and characterization, and lastly (4) foam properties and applications.