Graphene is nature's ideal two-dimensional conductor and is comprised of a single sheet of hexagonally packed carbon atoms. Since the first electrical measurements made on graphene, researchers have been trying to exploit the unique properties of this material for a variety of applications that span numerous scientific and engineering disciplines. In order to fully realize the potential of graphene, large scale synthesis of high quality graphene and the ability to control the electronic properties of this material on a nanometer length-scale are necessary and remain key challenges. This article will review the efforts at the Center for Nanoscale Materials that focus on the atomic-scale characterization and modification of graphene via scanning tunneling microscopy and its synthesis on various materials (SiC, Cu(111), Cu foil, etc.). These fundamental studies explore growth dynamics, film quality, and the role of defects. The chemical modification of graphene following exposure to atomic hydrogen will also be covered, while additional emphasis will be made on graphene's unique structural properties.