The purpose of this thesis is to investigate the electronic and structural properties of one- and two-dimensional materials such as graphene, graphene-like transition metal chalcogenides by using density functional theory. The single-atom thickness of graphene sheet is a novel material and attracts great interest due to its unique features. In recent years, theoretical and experimental studies on graphene provide quick knowledge and have opened up possibilities for many other two-dimensional new materials. Among these materials, especially transition metal chalcogenides have recently been the focus of studies of condensed matter physics. Unlike many superior properties of graphene, lack of band gap in electronic structure have highlighted the necessity of such transition metal chalcogenides materials for electronic applications. As compared to graphene, transition metal chalcogenides have various physical properties and possess sizable band gaps, for this reason they are promising candidate for many applications. Many experiments have revealed that the surfaces of graphene and graphene-like structures can play an active role as a host surface for clusterization of metal atoms. Motivated by these observations, we investigate characteristic properties of Pt atoms on graphene, MoS2 and TaS2. Similarly, TiSe2 is very recently synthesized two-dimensional transition metal dichalcogenide material and stable in 1T phase. Two-dimensional TiSe2 has a metallic electronic property and widely studied material. We analyze how to change the structural and electronic properties of TiSe2 by functionalization with hydrogen atom. Again to the effects of hydrogenation on two-dimensional TiSe2 monolayer we also study the structural and electronic properties of this material in nanoribbon form. At the same time, PtSe2 which is also very recently synthesized two-dimensional transition metal dichalcogenide and stable in 1T phase like TiSe2, its nanoribbon structural and electronic properties have also been investigated and compared with TiSe2 nanoribbons. Finally, TiS3 which is also transition metal chalcogenide but entirely different crystal structure, is recently widely studied materials. The structural and electronic properties as well as carrier mobility and strain response of TiS3 nanoribbons have been investigated. Besides many comprehensive theoretical studies, a lot of experimental studies are avaibale about the synthesis of these materials. In brief, these materials which tackles a contemporary and rapidly developing field, the nanoribbon form and functionalization of them that hold promise for many other applications.