In this article, I describe my early interest in graphene and contributions that I and my co-authors, in particular, have made to the field, along with a brief history of the experimental discovery of graphene. I then turn to new carbon materials whose experimental syntheses might be on the horizon. One example involves using graphene as a template to generate large-area ultrathin sp3-bonded carbon sheets that could also be substitutionally doped with, for example, nitrogen atoms, as one approach to making materials of interest for quantum computing. Such large-area sp3-bonded carbon sheets hold tremendous promise for use in thermal management; as a new material for electronics and photonics; and as ultrahigh-strength components in various structures including those used in aerospace, among other applications. Another example is the class of negative-curvature carbons (NCCs) that have atom-thick walls and carbon atoms trivalently bonded to other carbon atoms. Such NCCs have a nanoscale pore structure, atom-thick walls, and exceptionally high specific surface areas, and they fill three-dimensional space in ways that suggest their use as electrode materials for ultracapacitors and batteries, as adsorbents, as support material for catalysts, and for other applications.