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Graphene is a relatively new material with unique properties that holds promise for electronic applications. Since 2004, when the first graphene samples were intentionally fabricated, the worldwide research activities on graphene have literally exploded. Apart from physicists, also device engineers became interested in the new material and soon the prospects of graphene in electronics have been considered. For the most part, the early discussions on the potential of graphene had a prevailing positive mood, mainly based on the high carrier mobilities observed in this material. This has repeatedly led to very optimistic assessments of the potential of graphene transistors and to an underestimation of their problems. In this paper, we discuss the properties of graphene relevant for electronic applications, examine its advantages and problems, and summarize the state of the art of graphene transistors.