It is called "Third Party Insurance Cover" and in most countries it is a legal REQUIREMENT to have this cover on ALL cars being driven on public roiads.
It is normally sold to you as "Third Party, Fire and Theft" and covers for those 3 eventualities.
The insurance that covers injury to YOU and YOUR car if you are At Fault is called "Comprehensive Cover" - (this includes "Third Party, Fire and Theft" cover too).
Although generically, it is "third-party coverage" (rather than "first-party coverage" which applies to insurance on your own property), the proper name is liability coverage.
The collision portion of your Auto Insurance Policy would pay for damages to your own auto when you are the at fault party. The liability portion of your auto insurance pays for damages you cause to another party Insurance Plus
Your Home Insurance company will cover the damages. Nobody is liable for an act of Nature.
The secondary insurance cover both pays and co-pays of the primary insurance depending with the insurance company.
Liability insurance pays for someone else's damages if an accident is your fault but won't cover your vehicle. Full coverage provides liability insurance as above but will also cover your damages to your own vehicle in an accident regardless of whose at fault, as well as theft, fire, etc.
Liability coverage covers you if you are found liable or at fault for damages. In the case of auto insurance, for example, this coverage pays for damages you cause in an accident that is you fault. This coverage will NOT cover your car damage, however. Just the other drivers car, medical bills, & other property damage.
When this happens, your Insurance company pays for damages. If the accident is your fault, your insurance rates can go up.
It depends on where you are. In the US, you or your insurance pays first; then you collect from the person who hit you for the cost of the hospital bill, damages, and pain/suffering. Get a good lawyer to haggle with the insurance of the other person.
There is not deductible with liability insurance coverage. Liability pays the party who is not fault for their damages without a deductible. If you were at fault collision would pay for damages to your vehicle but you will have a deductible of whatever you selected when you purchased the insurance policy.
In most cases, when your car is damaged by a shopping cart, your own insurance company will cover the damages. Most shopping centers post "park at your own risk signs" but with the help of an attorney, you may get them to cover some of the repair cost.
Most times punitive damages are paid by the defendant and not by the defendant's insurance company. This is because insurance policies protect against accidental losses, like car accidents. Punitive damages are allowed only for intentional torts, which are not included in insurance coverage. If your neighbor falls down your stairs and gets hurt because you were negligent in some way, the insurance company pays for it. If you push him down the stairs, it will not cover the punitive damages you might get hit with.
It depends on the state, but at least where I live, the insurance of the stolen car pays for damages. Check it out with your insurance agent. He/she should be able to give you that information, even if your liability won't cover it.