A nonvolatile memory cell comprising a switchable resistor memory element and a thin-film three-terminal switching device, preferably a MOSFET, in series. The switchable resistor memory element has the property of having at least two stable resistance states, for example a high-resistance state and a low-resistance state. It is switched between the two states, and its resistance state (and thus the data state of the cell) is sensed by providing appropriate current through the three-terminal switching device. Preferred embodiments of the present invention include a highly dense monolithic three dimensional memory array in which multiple memory levels of such memory cells are formed above a single substrate such as a monocrystalline silicon wafer.