The isolation of graphene has triggered an avalanche of studies into the spin-dependent physical properties of this material and of graphene-based spintronic devices. Here, we review the experimental and theoretical state-of-art concerning spin injection and transport, defect-induced magnetic moments, spin-orbit coupling and spin relaxation in graphene. Future research in graphene spintronics will need to address the development of applications such as spin transistors and spin logic devices, as well as exotic physical properties including topological states and proximity-induced phenomena in graphene and other two-dimensional materials.