Potential applications of bulk GaN and AlN crystals have made possible single and multilayer allotropes of these III-V compounds to be a focus of interest recently. As of 2005, the theoretical studies have predicted that GaN and AlN can form two-dimensional (2D) stable, single-layer (SL) structures being wide band gap semiconductors and showing electronic and optical properties different from those of their bulk parents. Research on these 2D structures have gained importance with recent experimental studies achieving the growth of ultrathin 2D GaN and AlN on substrates. It is expected that these two materials will open an active field of research like graphene, silicene, and transition metal dichalcogenides. This topical review aims at the evaluation of previous experimental and theoretical works until 2018 in order to provide input for further research attempts in this field. To this end, starting from three-dimensional (3D) GaN and AlN crystals, we review 2D SL and multilayer (ML) structures, which were predicted to be stable in free-standing states. These are planar hexagonal (or honeycomb), tetragonal, and square-octagon structures. First, we discuss earlier results on dynamical and thermal stability of these SL structures, as well as the predicted mechanical properties. Next, their electronic and optical properties with and without the effect of strain are reviewed and compared with those of the 3D parent crystals. The formation of multilayers, hence prediction of new periodic layered structures and also tuning their physical properties with the number of layers are other critical subjects that have been actively studied and discussed here. In particular, an extensive analysis pertaining to the nature of perpendicular interlayer bonds causing planar GaN and AlN to buckle is presented. In view of the fact that SL GaN and AlN can be fabricated only on a substrate, the question of how the properties of free-standing, SL structures are affected if they are grown on a substrate is addressed. We also examine recent works treating the composite structures of GaN and AlN joined commensurately along their zigzag and armchair edges and forming heterostructures, δ-doping, single, and multiple quantum wells, as well as core/shell structures. Finally, outlooks and possible new research directions are briefly discussed. © 2018 Author(s).