The morphology of graphene is crucial for its applications, yet an adequate theory of its growth is lacking: It is either simplified to a phenomenological-continuum level or is overly detailed in atomistic simulations, which are often intractable. Here we put forward a comprehensive picture dubbed nanoreactor, which draws from ideas of step-flow crystal growth augmented by detailed first-principles calculations. As the carbon atoms migrate from the feedstock to catalyst to final graphene lattice, they go through a sequence of states whose energy levels can be computed and arranged into a step-by-step map. Analysis begins with the structure and energies of arbitrary edges to yield equilibrium island shapes. Then, it elucidates how the atoms dock at the edges and how they avoid forming defects. The sequence of atomic row assembly determines the kinetic anisotropy of growth, and consequently, graphene island morphology, explaining a number of experimental facts and suggesting how the growth product can further be improved. Finally, this analysis adds a useful perspective on the synthesis of carbon nanotubes and its essential distinction from graphene.