You’ll notice that none of that liability coverage pays for your car or injuries, nor for any injuries your passengers sustain if you cause a wreck. This is why many people — particularly those whose car isn’t yet paid off — want “full coverage” car insurance. This isn’t actually a type of coverage, but instead typically refers to policies that include liability coverage, plus comprehensive and collision coverages.
Minimal insurance policies cover only third parties (including the insured person and third parties carried with the vehicle, but not the driver, if the two do not coincide). Also the third parties, fire and theft are common insurance policies, while the all inclusive policies (kasko policy) which include also damages of the vehicle causing the accident or the injuries. It is also common to include a renounce clause of the insurance company to compensate the damages against the insured person in some cases (usually in case of DUI or other infringement of the law by the driver).
Several jurisdictions have experimented with a "pay-as-you-drive" insurance plan which utilizes either a tracking device in the vehicle or vehicle diagnostics. This would address issues of uninsured motorists by providing additional options and also charge based on the miles (kilometers) driven, which could theoretically increase the efficiency of the insurance, through streamlined collection.