This article presents an overview of the research highlights in graphene synthesis by CVD. The growth mechanisms over transition metals and alloys (with emphasis on Cu and Cu alloys) are discussed, including new developments and experiments in transfer-free graphene growth on dielectric materials. The focus is on the role of the various synthesis parameters, including the thermodynamic aspects of the chemical process, and the physical, chemical, and morphological properties of the substrate catalyst. The relationships between these parameters, and the properties of the as-grown graphene are discussed. Some important relationships are reviewed and used to influence the fundamental parameters of, and methods for, the synthesis of high quality graphene.