The semimetallic, linearly dispersive low-energy band structure of graphene provides a rich playground for physicists and a promising material for applications. Nonetheless, in many applications (e.g., solar energy harvesting) a gap on the eV scale is desirable. From a fundamental perspective, introducing a gap also allows for probing excitonic interactions that are otherwise obscured in extended graphene. Such gaps can be introduced by quantum confining graphene on the few-nanometer scale. Given that graphene is a two-dimensional lattice of light atoms, Coulombic interactions should be extremely strong and so essential to understanding the fundamental properties of electron-hole excitations and why graphene is even gapless. I will describe progress in synthesis and experimental and theoretical understanding of graphene quantum dots with well defined structures. I will highlight the importance of edge structure in understanding the properties of graphene quantum dots and the challenges in the synthesis of structures characterized by zero-energy states displaying magnetic ordering. I will emphasize recent studies of the excitonic properties of graphene quantum dots on the 2-3 nm scale and their connection to fundamental issues such as excitonic instability in extended graphene and practical issues such as carrier multiplication in quantum-dot-based photovoltaics.