A review. Carbon can exist in many different forms at different temps. and pressures. Some allotropes of carbon have been predicted in theory but still have not been found in nature. In this article, we mainly overview a no. of three-dimensional (3D) cryst. carbon allotropes, predicted by ab initio calcns. Particular attention will be placed on the carbon foams, which possess porous structures with a large surface area. Carbon foams are mostly composed of graphite segments connected by different types of carbon bonds. We will also review 3D carbon superstructures of low-dimensional allotropes, typically built from carbon fullerenes, nanobuds, nanotubes and graphene nanoribbons, as well as various other 3D cryst. carbon structures. Some of these carbon superstructures are composed of mixed sp-sp2 carbon or pure sp2 carbon (e.g., H-6, bct-4, C-20, K4), and some have larger mass d. than diamond (C8, hP3, tl12, tp12), and some can be transformed from graphite at room temp. and high pressure (e.g., M carbon, bct-4 carbon, W carbon, Z carbon). Some of these theor. predicted carbon allotropes may be synthesized in the lab. in future.