Graphene has a unique atom-thick two-dimensional structure and excellent properties, making it attractive for a variety of electrochemical applications, including electrosynthesis, electrochemical sensors or electrocatalysis, and energy conversion and storage. However, the electrochemistry of single-layer graphene has not yet been well understood, possibly due to the technical difficulties in handling individual graphene sheet. Here, we report the electrochemical behavior at single-layer graphene-based electrodes, comparing the basal plane of graphene to its edge. The graphene edge showed 4 orders of magnitude higher specific capacitance, much faster electron transfer rate and stronger electrocatalytic activity than those of graphene basal plane. A convergent diffusion effect was observed at the sub-nanometer thick graphene edge-electrode to accelerate the electrochemical reactions. Coupling with the high conductivity of a high-quality graphene basal plane, graphene edge is an ideal electrode for electrocatalysis and for the storage of capacitive charges.