Because of its unique properties and promising applications, graphene has attracted great interest from both academia and industry in the last decade. For studies on graphene as well as for applications, it is essential to develop techniques to prepare graphene in a controllable way. Graphene can be produced in the form of nano-/microflakes using a top-down method by the exfoliation of graphite or the reduction of graphene oxide, or in the form of a film or foam in a bottom-up method, predominantly by chemical vapor deposition of carbon precursors on catalytic substrates. This article focuses on the catalytic substrates, especially metals, used for graphene growth. We also discuss graphene growth mechanisms and kinetics, control of the number of graphene layers and their stacking order, engineering of large-area graphene single crystals, as well as low-temperature growth.