Two-dimensional (2D) nanomaterials have attracted increasing attention because of their unusual physical and chemical properties. Among these 2D nanomaterials, the monolayers of layered transition metal dichalcogenides exhibit intriguing physical and chemical properties. In contrast to the graphene, they are direct gap semiconductors with tunable band structures by controlling the composition, functionalizing, and applying external fields. In this review, the recent progress on the first-principles studies of MoS₂ is presented. The electronic and optical properties of MoS₂ monolayer, spin Hall Effect, lattice dynamics, functionalization, and H adsorption and diffusion, are systematically reviewed to provide a broad overview on its applications in electronics, optoelectronics, spintronics, sensor, and membrane. The recent advances on the MoS₂ nanoribbon, including edge stability, edge-dependent magnetism and electronic properties, and its application in renewable energy storage, are also presented. The first principles studies, supported by experimental results, show that their morphologies and properties are useful for nanodevices, catalysts, and energy storage applications.