When wisdom teeth erupt in an abnormal position due to lack of space, it's known as an impaction, and it can be nasty. Impacted teeth can grow in at an angle or even end up completely horizontal. Wisdom teeth can also break through partially, leaving a flap of gum over the top (which allows debris and food to become trapped inside). Sometimes the tooth never fully emerges, remaining wedged between the jawbone and the gum. Any of these scenarios can result in infections and damage to surrounding teeth, gums, bone and nerves. Sometimes a cyst forms around the tooth; left untreated, this can lead to tumors in rare cases.
Because of the potential for problems, many dentists recommend pre-emptive third molar removal. The American Academy of Oral and Maxillofacial Surgeons (AAOM) states that even if you don't feel pain, wisdom teeth can still cause damage, and leaving them could increase your risk of developing periodontal disease later in life. It strongly recommends that wisdom teeth be removed in young adults. The younger you are when they're removed, the easier the surgery will be, because your jawbone is softer and the teeth's roots haven't yet fully developed. Sometimes the roots can extend to nerves in the jaw, making the extraction more complicated. That's why most people who have their wisdom teeth removed do so between 16 and 18 years old.
The American Dental Association takes a more neutral stance, stating that "wisdom teeth are a valuable asset to the mouth when they are healthy and properly positioned" [source: American Dental Association]. The American Public Health Association is strongly against the removal of third molars -- even impacted ones, if there are no symptoms -- calling it "non-evidence based." Where your dentist went to school may also make a difference in his or her recommendation; for example, North American dentists usually recommend removal, while South American dentists are less likely to do so.
Wisdom tooth extraction surgery comes with risks of its own. In addition to the potential for complications from anesthesia, you can also experience temporary or permanent damage to nerves in the jaw, the joints or surrounding teeth. Recovering from the surgery typically means spending a few days of spring or winter break with a painful, swollen jaw; bleeding, oozing areas; and a diet of soft foods like Jell-O.
So the question isn't just when wisdom teeth should come out, but whether they should come out at all. The only person who can answer that question is the patient (and his or her parents), taking the advice of a dentist into consideration, of course.