Is an authorized user of your car has an accident but they are not on your policy, will your insurance company still cover you?

Your insurance will cover the driver drving your car even though they are not on your policy or the owner of your car.  However, this only happens in the event that the driver is at fault.  If the driver is at fault for the accident, the other not at fault driver will make a claim agains the driver and you.  most states have what is called permissive use, which holds the owner of the car equally liable as the driver that they let take their car.   If you can prove that the driver did not have permission, you can get your insurance company to deny the claim and the driver will be held liable for the damages.  if your insurance pays out the claim for the drvier's fault in an accident, the insurance company is going to then sue the driver personally to collect the money it paid.

An authorized user of the car, not listed on the insurance policy, will still be covered provided that they are not a household member. 

All the household residents, 16 (some states 15) years of age or older, and all regular operators of the vehicles need to be included on the policy.  You can choose to include the resident on your policy or not and companies' underwriting will determine if you are eligible to be insured by the insurance company.

If such a houshold member is not listed, your insurer will refuse coverage on them in the event of an accident.