Typically used in showers these valves automatically adjust the amount of cold and hot water being let through the valve to keep the output temperature more or less constant. The idea being that when someone flushes the toilet or starts the washing machine while you are in the shower you don't get burned. Having grown up without these great devices we were somewhat skeptical but extremely interested in seeing how they worked. When updating some plumbing in our house we decided to give them a try. After debating for a long time as to what sort to get we ended up going with the standard American Standard model (not the super fancy one). We've found that American Standard seems to produce very good quality plumbing fixtures but you certainly pay for the quality. After finally installing the valve and finishing the shower off we got our chance to test it. With the shower running we reached over and flushed the toilet... You could hear the valve make a quick clicking noise... And the water stayed the same temperature. What do you know? These things really do work.
Now the problem is that we've become so used to having these that when we travel we sometimes get a rude shock.