The answer varies, according to the type of insurance.
Under the Patient Protection and Affordable Care Act, the federal health care reform law, young adults up to age 26 can remain on their parents' health insurance policies if they don't have coverage through their own employers. Once you turn 26, you have to find your own health insurance. Keep in mind that under the federal health care reform's individual mandate, you'll be required to have coverage or face a tax penalty in 2014--unless the U.S. Supreme Court rules the mandate is unconstitutional. The court is expected to make a ruling by summer 2012.
As long as you live with your parents, your personal belongings are protected by their home insurance policy. Their home insurance also provides liability protection in case you damage someone else's property or injure someone and are sued.
Going off to college? Your parent's home insurance covers your belongings if you live on campus in a dorm. Beware there are limits on certain items, such as electronic equipment, so you might need additional coverage for those. As soon as you move into an apartment, though, whether you're attending college or working full-time, you need your own renters insurance policy. Of course you'll need your own condo or home insurance when you purchase a condo or house.
You can stay on your parents' car insurance policy as long as you live at home and they own the cars your drive. Once you purchase your own car, most car insurance companies require you to have your own policy, even if you still live at home with your folks.
Car insurance is pricey for teen-agers and young adults because they're at a higher risk for accidents than older adults. To learn how to save money on car insurance, see "Find cheap auto insurance in 8 easy steps."