A series of experiments is reported on rivalrous-texture stereograms composed of narrowband-filtered random noise. Experiment 1 found that the apparent deth-disparity function for such stereograms was different from that observed with similar but nonrivalrous stimuli. In particular, rivalrous divergent disparities produced the same depth as rivalrous zero disparity and this latter disparity itself produced a significant degree of protruding (i.e. 'convergent') depth in a certain type of rivalrous-texture stereogram. Free inspection was permitted and disparities were in the range 16 min convergent to 16 min divergent. Experiment 2 found no convincing evidence for reliable qualitative depth discriminations from tachistoscopic presentations of rivalrous-texture stereograms, using a forced-choice task requiring a discrimination between 16 min convergent and 16 min divergent conditions. This task was solved easily for equivalent nonrivalrous stimuli. Experiment 3 measured a hitherto unreported binocular depth effect, termed 'paradepth', which is produced by presenting a target in one field only. This effect appears to be a genuine biocular depth effect and not just the result of an ordinary monocular masking depth cue. The size of the depth effect was found to be a function of the width of the target. The overall conclusion derived from the series of experiments is that rivalrous-texture stereograms are complex stimuli capable of yielding curious and unexpected depth effects which are not readily explained in detail within any existing theoretical framework.