An optical method of fabricating randomly rough one-dimensional surfaces is described. The variations in the surface profile are produced by exposing photoresist-coated plates to a narrow line of light and scanning them under computer control. A theoretical analysis of the basic statistical properties of the fabricated surfaces is presented. These surfaces are in general non-Gaussian, but their statistics can be easily calculated, making them attractive for experimental and theoretical work. Several such surfaces have been fabricated and characterized with a stylus profilometer. The estimated statistical properties are in agreement with the theoretical predictions.