Nothing about divorce is ever totally black-and-white, including child support. Under most circumstances, you won't have to pay support if your children are over the age of 21. But exceptions exist to every rule, so it's possible that you might end up paying for a short while after your child's 21st birthday -- or even indefinitely.
Age of Majority
Each state recognizes a statutory age at which your child is legally considered an adult, and child support is no longer payable on her behalf after this time. In a majority of states, this age is 18. If your child is still in high school, some states require that you continue paying until she graduates. In a handful of states -- including New York, Indiana and Mississippi -- your child isn't legally an adult until she turns 21. If you divorce in one of these jurisdictions, you'll have to pay support through this time, unless she strikes out on her own. In this case, the law usually considers her self-supporting.
Some states require that divorced parents pay for their child's college costs. This may be in lieu of child support, or in addition to it, depending on where you live. Some jurisdictions, such as Colorado, cap college support at your child's 21st birthday, so even if you must pay it, you would not have to do so beyond age 21.
If your child is incapable of earning a living because she's mentally or physically disabled, her age doesn't matter. Support will continue indefinitely unless the court determines at some point that she's able to support herself.