What is my liability when my car was stolen and involved in a hit-and-run accident?
If the car was stolen, you are not liable for any accidents the criminal got into, since you were not "at fault." The criminal/driver did not have your permission to drive the car, so you are not responsible for what he/she did. That doesn't mean the other side may not try to sue you, but if you can prove theft, you'd have a good defense.
Legally, you are not liable for the car damages to the other person’s car (or bodily injury to a person) IF your car was stolen and involved in a hit-and-run traffic accident. Since you had nothing to do with the theft, you have no liability for what the thief did, and therefore do not have to spring for any bills incurred from the hit and run. Furthermore, since the car was stolen, your insurance company is not obligated to cover the damages caused by the actions of the thief either.
This flows out of two related legal principles. First, that a vehicle owner is liable for permitted uses of his or her car—and by definition, if the car was stolen, the use was not permitted. Second, that one person is not liable for the criminal acts of another unless the first person in some participated in or facilitated them.
That doesn't mean the other side may not try to sue you personally to repair the vehicle, since the courts do not “prescreen” lawsuits to see if they are valid or reasonable—almost anyone can initially file almost any lawsuit they want. However, the person suing you has the “burden of proof” to show you caused or were otherwise responsible for the crash. Simply owning the car is not enough to give rise to liability (since you have to be at fault in some way). Therefore, if you can prove theft, you'd have a good defense.
This means that proving that your car was stolen is critical. Hopefully the first thing you did was report your car as stolen to the police, giving them the make, model, license plate number, color, and year of the car along with the date and its last location. Filing a police report about car theft (and being willing to press charges, if the thief is caught) is strong evidence that this was theft. Send or fax a copy of the police report on file to your own insurance carrier, as well as to the other side (and/or its insurance company) if they are suing you.
A very slim chance, but the thief who caused the damage may have his own car insurance, assuming s/he can be identified and tracked down, and you can make a claim against the thief’s insurer. Too, if the thief is caught, you can file a civil lawsuit and get a judgment (though people without insurance are generally broke and don’t have much in assets to pay a judgment; and most people who steal cars do not have gainful employment).
In terms of repairing the damage to your car, you probably have to use your own collision or uninsured property damage coverage—assuming you have such insurance—to make the repairs, after the deductible. Your insurance company will fix (or total) your car and then go after the at-fault thief (assuming they are lucky enough to track him down). If you do not have collision coverage, you're on your own with regard to the damage to your car (especially if you can't find the thief/driver that hit it) and will unfortunately have to foot the expenses out of your own pocket.