Graphene, one of the most promising two-dimensional (2D) nanomaterials, has gained substantial attention in several areas of materials science. Due to its unique mechanical, electrical, optical, and thermal properties, graphene-based materials have triggered both numerous fundamental studies and technological applications. Out of several synthetic methods, chemical vapour deposition (CVD) has emerged as one of the most promising methods for the production of large areas of high quality single-crystal graphene. This review introduces the fundamental growth mechanisms of CVD graphene, alongside the various parameters and substrates employed in this process. Furthermore, new developments in the CVD synthesis of monolayer and few-layer graphene are presented, as well as advanced techniques for analysing the fine structure and properties of graphene. Moreover, a detailed discussion of the transfer processes used for practical applications of CVD graphene is provided, with emphasis on their fundamental aspects. This review concludes with an outlook on presently challenging issues, prospects and applications of graphene.