First-principles calculations based on density functional theory are employed to study and predict the properties of boron and Mg boride nanostructures. For boron nanostructures, two-dimensional boron sheets are found to be metallic and made of mixtures of triangles and hexagons which benefit from the balance of two-center bonding and three-center bonding. This unusual bonding in boron sheets results in a self-doping picture where adding atoms to the hexagon centers does not change the number of bonding states but merely increases the electron count. Boron sheets can be either flat or buckled depending on the ratio between hexagons and triangles. Formed by stacking two identical boron sheets, double-layered boron sheets can form interlayer bonds, and the most stable one is semiconducting. Built from single-layered boron sheets, single-walled boron nanotubes have smaller curvature energies than carbon nanotubes and undergo a metal-to-semiconductor transition once the diameter is smaller than ∼20 Å. Optimal double-walled boron nanotubes with inter-walled bonds formed are metallic and always more stable than single-walled ones. For Mg boride nanostructures, certain Mg boride sheets prefer to curve themselves into nanotubes, which is explained via Mg-Mg interactions governed by the charge state of Mg. In addition, optimal Mg boride sheet structures are explored with a genetic algorithm. Phase diagrams for Mg boride sheet structures are constructed and stable phases under boron-rich environments are identified. Curvature effects on the phase diagram of Mg boride nanotubes are also discussed. As a natural extension to boron sheets, layered boron crystals based on boron sheets are then presented and are shown to be stable under high pressure. Finally, this thesis ends with an investigation of hydrogen-storage properties of pristine and metal doped boron nanostructures.