Most states require motorists to have liability insurance to cover the cost of bodily injury and property damage to others. Each state normally mandates a minimum amount of coverage that you are required to have, and these minimum amounts vary by state (it is actually not normally possible to purchase an amount below the minimum requirements in your state).
However, there are a few exceptions:
The state of New Hampshire does not require motorists to carry liability insurance.
The state of Virginia does not require motorists to carry liability insurance, but if you do not obtain liability insurance you must pay an annual fee to the state of $500 per vehicle.
You do not normally need insurance to cover injuries to yourself or your own property. But if you lease a vehicle or purchase a vehicle on finance, the lease/finance agreement will often require you to maintain a current insurance policy for the vehicle. This is to protect the lessor/financer.