A method to form a rewriteable nonvolatile memory cell is disclosed, the cell comprising a steering element in series with a carbon nanotube fabric. The steering element is preferably a diode, but may also be a transistor. The carbon nanotube fabric reversibly changes resistivity when subjected to an appropriate electrical pulse. The different resistivity states of the carbon nanotube fabric can be sensed, and can correspond to distinct data states of the memory cell. A first memory level of such memory cells can be monolithically formed above a substrate, a second memory level monolithically formed above the first, and so on, forming a highly dense monolithic three dimensional memory array of stacked memory levels.