Graphene has been widely studied for its high in-plane charge carrier mobility and long spin diffusion lengths. In contrast, the out-of-plane charge and spin transport behavior of this atomically thin material have not been well addressed. We show here that while graphene exhibits metallic conductivity in-plane, it serves effectively as an insulator for transport perpendicular to the plane. We report fabrication of tunnel junctions using single-layer graphene between two ferromagnetic metal layers in a fully scalable photolithographic process. The transport occurs by quantum tunneling perpendicular to the graphene plane and preserves a net spin polarization of the current from the contact so that the structures exhibit tunneling magnetoresistance to 425 K. These results demonstrate that graphene can function as an effective tunnel barrier for both charge and spin-based devices and enable realization of more complex graphene-based devices for highly functional nanoscale circuits, such as tunnel transistors, nonvolatile magnetic memory, and reprogrammable spin logic.