Metal foils, particularly copper and copper-nickel alloy, are commonly used to grow large-area crystalline mono- or bi-layer graphene domains and films by chemical vapor deposition (CVD) methods. Thin-metal films, which are usually made by depositing metals on various substrates such as single-crystal sapphire, have also been reported as catalytic substrates for high-quality graphene growth. Thin-metal films can also serve as intermediates to grow graphene on catalytically inactive substrates, such as dielectrics for electronic devices. Focusing on the CVD growth of graphene on thin-metal films, we review the history of CVD graphene growth, the growth on different single-metals and alloy thin films, and the reported performance of such graphene in electronic devices. We also comment on current challenges and opportunities for the further development of this field.