The research on two-dimensional nanomaterials has grown exponentially after the experimental realization and characterization of graphene in 2005. Many new atom-thick material sheets have been theoretically proposed and experimentally realized since then and new possible structures are still predicted on a regular basis. To find new and possibly better materials for opto-electronics applications, it seems natural to investigate if 3D bulk materials that are already used for these applications can be scaled down to the 2D single layer limit, to even improve or tune their properties. This route has been followed for the group IV elements, leading to silicene, germanene, etc. Surprisingly, much less attention has been paid to the class of III-V materials, like InAs, GaAs, … Only the properties of these III-V compounds in the graphene-like (flat) or silicene-like structure (buckled) have thoroughly been investigated. However, it is well-known from molecular chemistry that group-III elements prefer planar sp2-bonded structures, as in trihydrides and trihalides, while group-V elements prefer tetragonal sp3-bonded configurations. It can be expected that these trends will pop-up again when reducing the 3D III-V bulk semiconductors to their 2D limit. The goal of this project is to identify the real ground state structures of 2D III-V compounds and to explore their electronic properties, using first-principles calculations. Another important field of research that arose from the research on graphene and graphene-like systems are topological insulators and the quantum spin Hall effect. The origin of this behavior lies in a band inversion, often realized by a strong spin-orbit coupling. So once we have determined the ground and metastable structures of the 2D compounds containing Tl and Bi, their topological character will be determined. The outstanding performance of many of today's opto-electronic devices is primarily due to the application of heterostructures, for example in lasers. Therefore it is also important to investigate the electronic properties of heterostructures composed of different 2D III-V semiconductors. Finally it is also natural to extend this research to the class of II-VI semiconductors.