Since the isolation of graphene and numerous demonstrations of its unique properties, the expectations for this material to be implemented in many future commercial applications have been enormous. However, to date, challenges still remain. One of the key challenges is the fabrication of graphene in a manner that satisfies processing requirements. While transfer of graphene can be used, this tends to damage or contaminate it, which degrades its performance. Hence, there is an important drive to grow graphene directly over a number of technologically important materials, viz., different substrate materials, so as to avoid the need for transfer. One of the more successful approaches to synthesis graphene is chemical vapor deposition (CVD), which is well established. Historically, transition metal substrates are used due to their catalytic properties. However, in recent years this has developed to include many nonmetal substrate systems. Moreover, both solid and molten substrate forms have also been demonstrated. In addition, the current trend to progress flexible devices has spurred interest in graphene growth directly over flexible materials surfaces. All these aspects are presented in this review which presents the developments in available substrates for graphene fabrication by CVD, with a focus primarily on large area graphene.