Graphene has changed from being the exclusive domain of condensed-matter physicists to being explored by those in the electron-device community. In particular, graphene-based transistors have developed rapidly and are now considered an option for post-silicon electronics. However, many details about the potential performance of graphene transistors in real applications remain unclear. Here I review the properties of graphene that are relevant to electron devices, discuss the trade-offs among these properties and examine their effects on the performance of graphene transistors in both logic and radiofrequency applications. I conclude that the excellent mobility of graphene may not, as is often assumed, be its most compelling feature from a device perspective. Rather, it may be the possibility of making devices with channels that are extremely thin that will allow graphene field-effect transistors to be scaled to shorter channel lengths and higher speeds without encountering the adverse short-channel effects that restrict the performance of existing devices. Outstanding challenges for graphene transistors include opening a sizeable and well-defined bandgap in graphene, making large-area graphene transistors that operate in the current-saturation regime and fabricating graphene nanoribbons with well-defined widths and clean edges.