I usually spay (female cats and dogs) and neuter (male cats and dogs) between 5 and 6 months of age.  At this point and time, most are not sexually mature, but are old enough to tolerate anaesthesia and surgery.

Many shelters in the US are spaying and neutering as early as 8 weeks of age.  This is done in order to ensure that all adopted pets leaving the shelters are sterilized and there is no chance of owners failing to follow through with spay/neuter agreements.