Weaving equipment may include warp strand positioning equipment that positions warp strands and weft strand positioning equipment that inserts weft strands among the warp strands to form fabric. The fabric may include insulating strands and conductive strands. Conductive strands may run orthogonal to each other and may cross at open circuit and short circuit intersections. The fabric may be formed using pairs of interwoven warp and weft strands. Conductive warp and weft strands may be interposed within the pairs of strands. The fabric may be a single layer fabric or may contain two or more layers. Stacked warp strands may be formed between pairs of adjacent insulating warp strands. The stacked warp strands may include insulating and conductive strands. Touch sensors and other components may include conductive structures that are formed from the conductive strands in the fabric.