If you are involved in an accident with a person who has no insurance and they are at fault does your insurance pay for repairs to your vehicle if you only have liability insurance?

No, Liability Insurance is just as it says, Liability only pays for damage you cause to another. It will not pay for damages to your vehicle if you had only liability coverage.

s from other contributors
Usually they don't. If you had no type of uninsured motorist protection which doesn't cost that much it's in your hands basically to repair your damages. You could sue the assailant for everything they are or ever will be worth but after you win the case they will eventually declare chapter 7 bankruptcy and you'll never get anything from them.
Take my suggestion and GET uninsured motorist coverage on your next vehicle. It may cost a little more but it's well worth it.
another answer:
I live in Texas, and, yes, the liability only pays for the other vehicle, assuming you were at fault. If the other driver was at fault, their insurance should pay you. HOWEVER, I want to explain something that most people don't know. If you have only liablility, and your car burns, you hit an animal, a tree falls on it, or ANYTHING that is not your fault, including running off the road and hitting something, you are just out of luck. BUT, if you have "COMPREHENSIVE INSURANCE" you will be covered for all of these things. Comprehensive insurance is VERY inexpensive, and if you have to file a claim using it, your rates will not increase. I drive older vehicles, and full coverage isn't feasable, so, I have liability and comprehensive on them. Several years ago, one of my trucks burned, and I was comphensated very fairly for it. If I had had only liability, well, I'd been walking. Please check it out if you are used to paying for only liability. I don't think comprehensive will pay if the other person just didn't have insurance, but, it will pay you if an "act of GOd" has happened, something unavoidable, like fire, and accidents not involving other vehicles.

In the state of Kentucky they have "no fault" insurance which is mandatory even on only liability....so you're covered in the event someone else causes the accident. However, after your car is fixed and there is no other insurance company to collect off, then your insurance will go up to pay for the claim. There is also "uninsured/under insured" riders. Check with your insurance company and state Dept of Insurance.