The nano-electronic technology has shown great potential as an innovative approach to incorporate various functionalities in molecular electronic devices. With this technology, many molecular devices have proposed functional properties such as molecular rectifying, negative differential resistance (NDR) and switch behaviors etc. Recently, carbon-based nano materials, such as carbon chains, carbon nanotube (CNT) and graphene, have attracted considerable attention in terms of experimental applications and theoretical research. At the same time, an applicable structure and modulation of their characteristics continue to elude researchers. This dissertation is motivated by the need to construct, validate and investigate transport properties of carbon-based molecular devices and systems theoretically. The first part of the thesis addresses introduction of silicon-based materials and their development. From this study, carbon-based materials are proposed as substitutes for silicon-based components, such as carbon chain, carbon nanotube and graphene which show more striking properties. These kinds of novel materials can be manufactured as nanometer-sized, free-standing, easily-modified, high stability and preferable molecular devices. The second part presents a simple overview of Born-Oppenheimer approximation, density-functional theory (DFT) and non-equilibrium Green's function (NEGF) which is the theoretical framework of our simulations. The method we used is called "first-principles" that is proven to be reliable and suitable for many nanoscale problems and its results can be compared with experimental analyses. The investigation of a single carbon atom doping-effect on transport properties of boron-nitride nanowires and the results show that this simple carbon-doping modulation can enhance functionality of BN nanowires with increase in the length of devices. The carbon chain- and monoatomic ring-based molecular devices sandwiched with Au electrodes based on first-principles are studied. For carbon chain-based molecular devices, transport properties obviously have odd-even effect of rectifying performance with length increasing. The rectifying ratios and NDR behaviors can be tuned by the degree of ring deformation for ring based molecular devices. They have obvious rectifying performance and NDR behaviors, and have the potential to be multifunctional molecular devices. The next focus of the thesis is on transport properties of simple layer graphene nanoribbon (GNR). In the first part, a single layer armchair h-BNC heterostructure is simulated while the GNR and boron-nitride nanoribbon (BNNR) are connected with the interface C-B bond with a certain proportion of elements. This heterostructure has been confirmed to be metastable and predicted to be a promising system for fundamental physical investigations. Results show it to be applicable and functional for multifunctional molecular devices. For this device, we validate the different vacancy positions effect on transport behaviors. The results show that NDR behaviors can be strengthened clearly with vacancy atoms near the interface of GNR and BNNR, and rectifying performance can be enhanced obviously when there are vacancy atoms in the moiety of BNNR. The feasibility of all-carbon-based systems is examined and validated. The rectifying performance and switching behaviors of nanoribbon-chain-CNT and sulfur terminated armchair graphene nanoribbon junctions are reported. In the first part, electronic transport properties of armchair graphene nanoribbon and capped carbon nanotube junctions, covalently bridged by carbon atomic chains with different numbers of carbon atoms, are investigated. The first-principles calculations show that their I-V characteristics display odd-even effects and forward rectifying behaviors show obvious oscillations. The analysis for even- and odd- numbered carbon chains affected contact bonds, charge transfer, density of states and evolutions of molecular orbitals which have reveals the intrinsic origins of the rectifying behaviors oscillated. In the second part, transport properties armchair graphene nanoribbon junctions of different widths having zigzag edges terminating with sulfur atoms are investigated. The calculations show that their I-V characteristics display obvious rectifying performance and switching behaviors, namely, different contact points, effects on contact bonds, surface potential, charge transfer, rectifying ratios, switch ratios and density of states. Our work predicts that this simple transport system, driven by micro-mechanical operations and/or thermal activations, has the potential for application as a multi-functional molecular modulator. In order to develop functionalization of multilayers graphene (graphite), graphite-chain-graphene nanoribbon junctions are constructed to study the striking properties of graphite deeply. The interesting rectifying performance and NDR behavior show many advantages over metal-based or silicon-based systems. The current in odd numbered chains is carried by resonances originating from hybridization of surface of graphite electrode, chain and nanoribbon states, while others are mediated by tunneling between graphite electrode and chain states. The rectifying performance in our models was found to be a result of asymmetric distortions of conducting resonances of the graphite electrode and chain states. It was also found that odd numbered chains exhibit NDR behaviors caused by depressed chain states. This paves a new way, from silicon-based components to a carbon-based era. Finally, challenges of the future works arising in quantum transport field are investigated through a critical appraisal.