If you can insure a car without registering it will vary from insurance company to insurance company since underwriting guidelines differ as do state laws
In general, insurance companies will request a current registration for three (3) reasons. First, it helps prove someone has "insurable interest" in a vehicle. A person has an insurable interest in something when loss or damage to it would cause that person to suffer a financial loss or certain other kinds of losses.
Second, some insurance companies require a vehicle be in a state a set period of time (such as 10 months). Registering a car with the state's Department of Motor Vehicle helps prove that the car is used and garaged within that state.
Third, there are usually state laws that require vehicles to be registered. There may be exceptions for project cars or cars that are being stored that you want insurance on but will have taken registration off until it goes back on the road. Again state laws regarding these types of situations differ in each state as to what is acceptable.
If you have a car that you want to insure without registering you can check with various insurance companies to see if their guidelines allow for coverage to be placed on the vehicle or not. It may depend upon the reason the car is not being registered but will always adhere to what state law first and foremost dictates and then the own guidelines of the individual insurance company.
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