A number of photomultiplier tubes have been assessed for application in experiments where the counting of individual photoelectrons from the photocathode is necessary or advantageous. Pulse height distributions, signal-to-noise-in-signal ratios, over-all quantum-counting efficiencies, time dependent statistical correlations, and dark current properties have been investigated and compared with theoretical expectations. Amajor finding has been the general low value of over-all quantum-counting efficiency. Direct measurements of this figure have not, to our knowledge, been published previously. A second conclusion has been that, although there seems to be no reason why high performance with respect to each of the features considered should not be achieved in a single tube, we have not yet found one in which this is so.