Graphene shows promise as a future material for nanoelectronics owing to its compatibility with industry-standard lithographic processing, electron mobilities up to 150 times greater than Si and a thermal conductivity twice that of diamond. The electronic structure of graphene nanoribbons (GNRs) and quantum dots (GQDs) has been predicted to depend sensitively on the crystallographic orientation of their edges; however, the influence of edge structure has not been verified experimentally. Here, we use tunnelling spectroscopy to show that the electronic structure of GNRs and GQDs with 2-20 nm lateral dimensions varies on the basis of the graphene edge lattice symmetry. Predominantly zigzag-edge GQDs with 7-8 nm average dimensions are metallic owing to the presence of zigzag edge states. GNRs with a higher fraction of zigzag edges exhibit a smaller energy gap than a predominantly armchair-edge ribbon of similar width, and the magnitudes of the measured GNR energy gaps agree with recent theoretical calculations.