We review the emerging field of organic spintronics, where organic materials are applied as a medium to transport and control spin-polarized signals. The contacts for injecting and detecting spins are formed by ferromagnetic metals, oxides, or inorganic semiconductors. First, the basic concepts of spintronics and organic electronics are addressed, and phenomena which are in particular relevant for organic spintronics are highlighted. Experiments using different organic materials, including carbon nanotubes, organic thin films, self-assembled monolayers and single molecules are then reviewed. Observed magnetoresistance points toward successful spin injection and detection, but spurious magnetoresistance effects can easily be confused with spin accumulation. A few studies report long spin relaxation times and lengths, which forms a promising basis for further research. We conclude with discussing outstanding questions and problems.