The effects of a magnetic field on the current in sandwich devices of a nonmagnetic material in-between two ferromagnetic electrodes are well known. However, magnetic-field effects also occur in the responses of devices of organic semiconductors sandwiched in-between non-ferromagnetic electrodes, providing an entirely new route toward organic spintronics. The precise origins of these "intrinsic" magnetic field effects are still unclear. They appear to be related to spin-selective reactions between paramagnetic entities such as electrons, holes, and triplet excitons. We present an overview of these effects and discuss three recent developments that shed new light on them: (1) tuning of the effects in molecularly engineered systems, (2) the discovery of ultrahigh magnetoresistance in molecular wires, and (3) the discovery of "fringe-field" magnetoresistance.