Low-dimensional materials including both 1D and 2D scenarios exhibit unique properties distinguished from their bulk states. In this thesis, computational modeling of low-dimensional materials on their structures, properties and applications has been investigated. First-principles simulations are employed to investigate the following topics. First of all in the 1D scenario, a comprehensive study on carbyne-one dimensional carbon chain-from its structure to properties has been conducted, and the extreme mechanical performance and intriguing metal-insulator transition under tension has been demonstrated. The properties of proposed 1D boron nanostructures have also been investigated and a constant-tension structural transition between two boron phases has been revealed. Secondly, two examples for the energy application of low-dimensional materials have been presented. The first example contains the energy storage with graphene and its derivatives applied in Li-ion batteries as well as the examination on the lithium nucleation process on graphene. The second example is the exploration of the energy conversion with N-doped carbon materials as effective catalysts in electrochemical reduction of CO2. Lastly, the simplified model- jellium model- has been applied in carbon nanotube growth. The termination effect and the chiral selectivity in CNT growth have been investigated.