A monolithic, three-dimensional memory device, comprising: a substrate having a major surface; a plurality of electrically conductive word lines over the major surface of the substrate, wherein each word line is elongated in a first direction that is substantially parallel to the major surface of the substrate and the plurality of word lines are spaced apart from one another in a second direction that is substantially perpendicular to the major surface of the substrate, and an electrically insulating material is located between each of the spaced apart word lines; an electrically conductive bit line extending in a direction substantially perpendicular to the major surface of the substrate and adjacent to each of the plurality of word lines; a non-volatile memory element material located between the bit line and each of the plurality of word lines; a plurality of middle electrodes comprising an electrically conductive material located between the bit line and each of the plurality of word lines, wherein the plurality of middle electrodes are discrete electrodes which are isolated from one another in at least the second direction; wherein the bit line comprises a generally pillar-shaped structure having at least one side surface, and the non-volatile memory element material comprises a continuous layer that extends over a side surface of the bit line and adjacent to each of the plurality of word lines and a non-linear element located between the bit line and each of the plurality of word lines, and characterized in that: the layer of non-volatile memory element material comprises a plurality of clam shape portions adjacent to each of the plurality of word lines; each of the clam shape portions surrounds a respective one of the plurality of middle electrodes on three sides; the bit line comprises the generally pillar-shaped structure having a substantially rectangular cross section; the non-linear element is located adjacent to all four sidewalls of the substantially-pillar shaped bit line; and the plurality of middle electrodes are located adjacent to two of four sidewalls of the substantially-pillar shaped bit lines. a substrate having a major surface; a plurality of electrically conductive word lines over the major surface of the substrate, wherein each word line is elongated in a first direction that is substantially parallel to the major surface of the substrate and the plurality of word lines are spaced apart from one another in a second direction that is substantially perpendicular to the major surface of the substrate, and an electrically insulating material is located between each of the spaced apart word lines; an electrically conductive bit line extending in a direction substantially perpendicular to the major surface of the substrate and adjacent to each of the plurality of word lines; a non-volatile memory element material located between the bit line and each of the plurality of word lines; a plurality of middle electrodes comprising an electrically conductive material located between the bit line and each of the plurality of word lines, wherein the plurality of middle electrodes are discrete electrodes which are isolated from one another in at least the second direction; wherein the bit line comprises a generally pillar-shaped structure having at least one side surface, and the non-volatile memory element material comprises a continuous layer that extends over a side surface of the bit line and adjacent to each of the plurality of word lines and a non-linear element located between the bit line and each of the plurality of word lines, and characterized in that: the layer of non-volatile memory element material comprises a plurality of clam shape portions adjacent to each of the plurality of word lines; each of the clam shape portions surrounds a respective one of the plurality of middle electrodes on three sides; the bit line comprises the generally pillar-shaped structure having a substantially rectangular cross section; the non-linear element is located adjacent to all four sidewalls of the substantially-pillar shaped bit line; and the plurality of middle electrodes are located adjacent to two of four sidewalls of the substantially-pillar shaped bit lines. the layer of non-volatile memory element material comprises a plurality of clam shape portions adjacent to each of the plurality of word lines; each of the clam shape portions surrounds a respective one of the plurality of middle electrodes on three sides; the bit line comprises the generally pillar-shaped structure having a substantially rectangular cross section; the non-linear element is located adjacent to all four sidewalls of the substantially-pillar shaped bit line; and the plurality of middle electrodes are located adjacent to two of four sidewalls of the substantially-pillar shaped bit lines.