Gas counters made from parallel metallic wire meshes have been proposed by Charpak et al. (1978) long ago as a preamplifying stage to wire chambers and developed meanwhile by many workers. The concept led to very robust and versatile detectors, having evolved over time to include complex multistep structures. It is known that parallel geometry chambers can stand counting rates at least up to 105 Hz/mm/sup 2/ without any pulse-height degradation, suggesting their use as high-rate particle detectors. In this work we describe a study of the high-rate behavior of parallel mesh chambers, measuring the maximum count rates achievable for a given avalanche charge and beam cross-section. The different breakdown regimes that appear at high counting rates are described.