A tunneling ferrimagnetic magnetoresistive sensor (10, 30, 110) that has a - R/R greater than that of known magnetoresistive sensors, and that, with appropriate electrode materials, can undergo a substantial change in resistance in response to a magnetic field in the intensity range of 10s of Oe, which is typical of the intensity of the magnetic fields encountered in magnetic recording media such as discs and tapes. The tunneling ferrimagnetic magnetoresistive sensor is composed of a stack of thin-film layers that include a layer (12) of a ferrimagnetic material, a layer (14) of a magnetic material, and a layer (16) of an insulator interposed between the layer of the ferrimagnetic material and the layer of the magnetic material. The ferrimagnetic material is conductive. The magnetic material is also conductive and has a coercivity substantially different from that of the ferrimagnetic material. The insulating layer is of a thickness that is sufficiently small to permit tunneling of current carriers between the layer of the ferrimagnetic material and the layer of the magnetic material.