(In the UK)
I'm guessing you mean on your own vehicle? i.e. your car repair bill is for £5000, but your vehicle is only worth £3000 (amounts for illustration only)?
In that situation, the Insurer would 'write off' the vehicle - essentially declaring it as unworthy of repair. They would then pay out to you the £3000, assuming that £3k is their valuation of the car and take ownership of it themselves for scrapping. This is regardless of what you told them it was worth when you took out the insurance. Conversely, if you told them it was worth £2500, and the car is worth £3000, they will only pay up to a maximum of £2500 - sneaky so & so's!!!
If you feel that the damage is only cosmetic, or you are a mechanic yourself, and feel that the car can be repaired cheaply, then you can always buy the car back off them - usually for a couple of hundred quid.
If however you meant other people's vehicles, then I believe it is the same as public liability insurance, and covers up to £1 million. Should be enough for most vehicles...
Outside of the UK I'm afraid I don't have a clue!
ANSWER
In the US, if the vehicle were itself damaged, once the deductible is paid by the insured, the insurer pays the reasonable cost of repair. Recently, it has become common for insurers to have preferred repair facilities that work with the insurer on cost of repairs. This is similar to managed care in the health insurance arena.
In most states, it is also the law that if damages exceed a stated percentage of the actual cash value of the car, the insured must declare it a total loss. The insurer then pays the insured the actual cash value, less the policy's collision or comprehensive deductible. The insured if often given a choice to either keep the salvage and, perhaps, pay to rebuild the vehicle, or to sign title over to the insurer. If the insurer keeps the salvage, the salvage value is deducted from the amount paid to the insured.
In cases where the insured is At Fault for a collision and a claim is made by a third party for property or bodily injury damage, most states require the insurer to settle the claim within the insured's policy limits if it possible to do so. If it unwilling to pay the value of the claim when it knew or should have known that the claim was in excess of policy limits, and the claimant will settle for policy limits, the insurer will have exposed the insurer to an "excess verdict". That is, in most states, considered "bad faith". In that situation, the insured can sue the insured for additional damages. Alternatively, the insured can assign that claim to the claimant who may sue the insurer for the difference between policy limits and the amount of the claim.
Note that state law governing bad faith may differ depending the jurisdiction, and that this is intended only as a general discussion.
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