This dissertation research is focused on first principles studies of graphene and single organic molecules for nanoelectronics applications. These nanosized objects attracted considerable interest from the scientific community due to their promise to serve as building blocks of nanoelectronic devices with low power consumption, high stability, rich functionality, scalability, and unique potentials for device integration. Both graphene electronics and molecular electronics pursue the same goal by using two different approaches: top-down approach for graphene devices scaling to smaller and smaller dimensions, and bottom-up approach for single molecule devices. One of the goals of this PhD research is to apply first-principles density functional theory (DFT) to study graphene/metal and molecule/metal contacts at atomic level. In addition, the DFT-based approach allowed us to predict the electronic characteristics of single molecular devices. The ideal and defective graphene/metal interfaces in weak and strong coupling regimes were systematically studied to aid experimentalists in understanding graphene growth. In addition, a theory of resonant charge transport in molecular tunnel junctions has been developed. The first part of this dissertation is devoted to the study of atomic, electronic, electric, and thermal properties of molecular tunnel junctions. After describing the model and justifying the approximations that have been made, the theory of resonant charge transport is introduced to explain the nature of current rectification within a chemically asymmetric molecule. The interaction of the tunneling charges (electrons and holes) with the electron density of the metal electrodes, which in classical physics is described using the notion of an image potential, are taken into account at the quantum-mechanical level within the tight binding formalism. The amount of energy released onto a molecule by tunneling electrons and holes in the form of thermal vibration excitations is related to the reorganization energy of the molecule, which is also responsible for an effective broadening of molecular levels. It was also predicted that due to the asymmetry of electron and hole resonant energy levels with respect to the Fermi energy of the electrodes, the Joule heating released from the metallic electrodes is also non-symmetric and can be used for the experimental determination of the type of charge carriers contributing to the molecular conductance. In the second part of the dissertation research ideal and defective graphene/metal interfaces are studied in weak and strong interface coupling regimes. The theoretical predictions suggest that the interface coupling may be controlled by depositing an extra metallic layer on top of the graphene. DFT calculations were performed to evaluate the stability of a surface nickel carbide, and to study graphene/carbide phase coexistence at initial stages of graphene growth on Ni(111) substrate at low growth temperatures. Point defects in graphene were also investigated by DFT, which showed that the defect formation energy is reduced due to interfacial interactions with the substrate, the effect being more pronounced in chemisorbed graphene on Ni(111) substrate than in physisorbed graphene on Cu(111) substrate. Our findings are correlated with recent experiments that demonstrated the local etching of transfered graphene by metal substrate imperfections. Both graphene and molecular electronics components of the PhD dissertation research were conducted in close collaboration with several experimental groups at the University of South Florida, Brookhaven National Laboratory, University of Chicago, and Arizona State University.