In the present thesis several aspects of graphene-based structures have been investigated using density functional theory calculations to solve the electronic structure problem. A review of the implementation of a localized basis-set within the projector augmented wave method - the way of describing the core electrons employed - is also presented. The investigation of the binding of graphene on metallic model surfaces is presented comparing the results from traditional exchange and correlation functionals to the results obtained with a new type of non-local functional, which includes van der Waals interactions. Relevant comparisons to experimental data are also pointed out. The stability of GNRs in the presence of common gas species adsorbed at the edge is also investigated. It is shown that the saturation of the edges by oxygen atoms is also important to consider, besides the well studied hydrogen passivation. A joint experimental and theoretical study of the mechanism by which suspended graphene is etched by catalytically active silver nanoparticles have been studied. The experimental observation of zigzag channels is elucidated by the DFT calculations, which show that the armchair edges are easier to remove and therefore only zigzag edges are left. Finally, functionalized graphene has been investigated as catalyst for the electrochemical reduction of CO2 to chemical fuels and comparisons are made with traditional transition-metal surfaces. The investigated porphyrin-like structures are attractive candidates although issues regarding the poisoning of the active site remain to be addressed.