if my car is hit by a hit & run driver, can I claim the damages on my auto insurance?

Yes, If you have collision coverage on your car, then you will be able to file a claim for the damages to your car. Of course your collision deductible will apply to this type of claim. People often think that they can file this type of claim under the uninsured motorists coverage but to do so, you will have to find the person who hit you and then prove that they don't have insurance.