The non-trivial properties of low-dimensional (LD) materials due to quantum confinement effect miraculously advance our understanding of the fundamental physics principles of nature. This thesis explores the electronic and mechanical properties, morphologies, synthesis mechanism, and electro-catalytic applications of LD materials through employing state-of-art computational method of first principles calculations, especially density functional theory (DFT). Regarding electronic and mechanical properties, band gap transitions of MoS2 and WS2, anisotropy of phosphorene and low contact barrier of 2H/1T’ MoTe2 junction were analyzed. About morphologies, strategies for asymmetric two-dimensional (2D) materials were proposed. With respect to synthesis, growth mechanism of borophene on Au (111) and line defects of that on Ag (111) are discussed. For electro-catalytic applications, hydrogen evolution reaction (HER), oxygen evolution reaction (OER) and oxygen reduction reaction (ORR) in 2D materials are illustrated. These theoretical investigations agree well with experimental measurements, and provide insights for the understanding of the mechanisms of LD materials.