Three-dimensional (3D) Gallium Nitride (GaN) is a III-V compound semiconductor with direct band gap. It is widely used in light emitting diodes (LED) and has potential to be used numerous optoelectronic applications. In this thesis, rstly 3D GaN in wurtzite and zincblende structures are revisited and structural, mechanical, and electronic properties are studied and compared with the literature. Next, the mechanical and electronic properties of two-dimensional (2D) single-layer honeycomb structure of GaN (g-GaN), its bilayer, trilayer and multilayer van der Waals solids are investigated using density functional theory. Based on phonon spectrum analysis and high temperature ab initio molecular dynamics calculations, rst it is showed that g-GaN is stable and can preserve its geometry even at high temperatures. Then a comparative study is performed to reveal how the physical properties vary with dimensionality. While 3D GaN is a direct band gap semiconductor, g-GaN in 2D has relatively wider indirect band gap. Moreover, 2D g-GaN displays higher Poisson's ratio and slightly less charge transfer from cation to anion. It is also showed that the physical properties predicted for freestanding g-GaN are preserved when g-GaN is grown on metallic, as well as semiconducting substrates. In particular, 3D layered blue phosphorus being nearly lattice matched to g-GaN is found to be an excellent substrate for growing g-GaN. Bilayer, trilayer and van der Waals crystals can be constructed by special stacking sequence of g-GaN and they can display electronic properties which can be controlled by the number of g-GaN layers. In particular, their fundamental band gap decreases and changes from indirect to direct with increasing number of g-GaN layers. It is hoped that the present work will provide helpful insights for growing g-GaN which can be widely used in nanoelectronics applications in low dimensions.