Two-dimensional Nanomaterials have been demonstrated to show superior properties and promising potential for applications. In this work, we investigate the structures and mechanical properties of several two-dimensional nanomaterials and their derivatives using various computational simulation methods, including boron, carbon nanotube, graphene, and boron nitride. The first part of the thesis focuses on the boron nanostructures. We report a comprehensive first-principles study of the structural and chemical properties of the recently discovered B40 cage. We also discover here a preferred structure of two-dimensional boron using the cluster expansion method and find it to be most table on reactive Cu and Ni. In the second part, an extensive analysis of the graphene grain boundaries is conducted and it is revealed that the sinuous grain boundaries based on dislocation theory and first-principles calculations can be energetically optimal once the global grain boundary line cannot bisect the tilt angle. In addition, we demonstrate here a contrasting behavior for grain boundaries in hybrid two-dimensional materials, which tend to be non-bisector and obey a universal law to optimally match the heterogeneous grains. In the last part, we propose an approach for determining the Gaussian bending modulus of graphene by utilizing carbon torus, whose topology enables its bending energy to be extracted from the coupled in-plane strain energy. Furthermore, we report a unique method to locally determine the mechanical response of individual covalent junctions between carbon nanotubes. Targeted synthesis of desired junction geometries can therefore provide a “structural alphabet” for construction of macroscopic carbon nanotube networks with tunable mechanical response.