Graphene is the name given to a monolayer of carbon atoms that are tightly packed into a two-dimensional honeycomb lattice. Since experimentally isolated in 2004, graphene has attracted widespread research interests for its outstanding electrical, optical, mechanical and thermal properties. The main theme of my doctoral research is to investigate the properties and potential applications of this magic material in electrochemistry and spintronics. In the first study, we investigated the electrochemistry of individual monolayer graphene. In particular, we fabricated devices in which a well-defined area of monolayer graphene served as the working electrode in electrochemical experiments. We examined both mechanically exfoliated graphene and graphene grown by chemical vapor deposition, with a focus on the interaction between graphene and the simple redox molecule ferrocenemethanol. We found the electron transfer rate of ferrocenemethanol at both types of graphene electrodes to be more than 10-fold faster than that at the basal plane of graphite, which we ascribed to corrugations in the graphene surfaces. We further demonstrated that molecule adsorption and desorption on graphene surface can be detected in real-time with electrochemistry techniques. In the second study, we explored the potential applications of graphene as a barrier material in magnetic tunnel junctions (MTJs). We developed a novel fabrication process that aimed to make graphene-based magnetic junctions with no oxidation of the ferromagnetic material (FM) at the FM/graphene interfaces. The oxidation issue was hard to avoid in traditional graphene fabrication processes, and may have accounted for the extremely large resistance-area (RA) products in previously reported graphene-based MTJs. Electrical characterization of our junction devices indicates that the junctions are tunnel junctions with 1-4% magnetoresistance and low RA products of 3-10 Ω[MICRO SIGN]m2. These RA values are ~104 times smaller than previous results, suggestive of intrinsic FM/graphene interfaces without oxidation.