The unique properties of graphene have led to the use of this allotrope of carbon in a wide range of applications, including semiconductors, energy devices, diffusion barriers, heat spreaders, and protective overcoats. The synthesis of graphene by process methods that either directly or indirectly rely on physical vapor deposition, thermal annealing, laser irradiation, and ion/electron beam irradiation has drawn significant attention in recent years, mainly because they can provide high purity, low temperature, high throughput, and controllable growth of graphene on various substrates. This article provides a comprehensive assessment of these methods by grouping them into two main categories, i.e., indirect methods in which a carbon layer is first deposited on a substrate and then converted to graphene by some type of energetic post-treatment process and direct methods in which graphene is directly synthesized on a substrate surface by a process that uses a solid carbon source. The underlying growth mechanisms of these processes and the challenging issues that need to be overcome before further advances in graphene synthesis can occur are interpreted in the context of published results.