Lateral and vertical heterostructures constructed of two-dimensional (2D) single-layer h-GaN and h-AlN display novel electronic and optical properties and diverse quantum structures to be utilized in 2D device applications. Lateral heterostructures formed by periodically repeating narrow h-GaN and h-AlN stripes, which are joined commensurately along their armchair edges, behave as composite semiconducting materials. Direct-indirect characters of the fundamental band gaps and their values vary with the widths of these stripes. However, for relatively wider stripes, electronic states are confined in different stripes and make a semiconductor-semiconductor junction with normal band alignment. This way one-dimensinonal multiple quantum well structures can be generated with electrons and holes confined to h-GaN stripes. Vertical heterostructures formed by thin stacks of h-GaN and h-AlN are composite semiconductors with a tunable fundamental band gap. However, depending on the stacking sequence and number of constituent sheets in the stacks, the vertical heterostructure can transform into a junction, which displays staggered band alignment with electrons and holes separated in different stacks. The weak bonds between the cations and anions in adjacent layers distinguish these heterostructures from those fabricated using thin films of GaN and AlN thin films in wurtzite structure, as well as from van der Waals solids. Despite the complexities due to confinement effects and charge transfer across the interface, the band diagram of the heterostructures in the direct space and band lineup are conveniently revealed from the electronic structure projected to the atoms or layers. Prominent features in the optical spectra of the lateral composite structures are observed within the limits of those of 2D parent constituents; however, significant deviations from pristine 2D constituents are observed for vertical heterostructures. Important dimensionality effects are revealed in the lateral and vertical heterostructures.