We predict that specific group-IV elements and IV-IV, III-V, and II-VI compounds can form stable, freestanding two-dimensional (2D) monolayers consisting of octagon, hexagon, and square rings (ohs), in which the threefold coordination of atoms is preserved to allow sp2-type hybridization. These monolayers can also construct bilayers, multilayers, three-dimensional (3D) layered van der Waals solids, and 3D crystals with strong vertical bonds between layers as well as quasi-one-dimensional nanotubes and nanoribbons with diverse edge geometries. All these ohs structures can constitute a large class of 2D materials ranging from good metals to wide bandgap semiconductors and display physical and chemical properties rather different from those of their counterparts in the hexagonal (honeycomb) network. The metallic state of freestanding 2D C, Si, and Ge ohs monolayers and 3D C ohs bulk contrast, respectively, with graphene, silicene, germanene, and graphite.