Because of its fascinating electronic properties, graphene is expected to produce breakthroughs in many areas of nanoelectronics. For spintronics, its key advantage is the expected long spin lifetime, combined with its large electron velocity. In this article, we review recent theoretical and experimental results showing that graphene could be the long-awaited platform for spintronics. A critical parameter for both characterization and devices is the resistance of the contact between the electrodes and the graphene, which must be large enough to prevent quenching of the induced spin polarization but small enough to allow for the detection of this polarization. Spin diffusion lengths in the 100-μm range, much longer than those in conventional metals and semiconductors, have been observed. This could be a unique advantage for several concepts of spintronic devices, particularly for the implementation of complex architectures or logic circuits in which information is coded by pure spin currents.