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Graphene, the wonder material of the 21st century, is expected to play an important role in future nanoelectronic applications, but the only way to achieve this goal is to grow graphene directly on a semiconductor, integrating it in the chain for the production of electronic circuits and devices. This book summarizes the latest achievements in this field, with particular attention to the graphitization of SiC. Through high-temperature annealing in a controlled environment, it is possible to decompose the topmost SiC layers, obtaining quasi-ideal graphene by Si sublimation with record electronic mobilities, while selective growth on patterned structures makes possible the opening of a gap by quantum confinement. The book starts with a review chapter on the significance and challenges of graphene growth on semiconductors, followed by three chapters dedicated to an up-to-date analysis of the synthesis of graphene in ultrahigh vacuum, and concludes with two chapters discussing possible ways of tailoring the electronic band structure of epitaxial graphene by atomic intercalation and of creating a gap by the growth of templated graphene nanostructures.