No, not all states require you drivers to buy liability insurance, but a majority of them do. The common question is, How much car insurance do I have to buy? All 50 states have different requirements when it comes to liability auto insurance and minimum insurance requirements. Almost every state requires you to have bodily injury liability insurance and every states has financial responsibility laws that require you be able to have sufficient assets to pay for any liability you cause in an incident.
In some states, drivers can't register a car without showing proof that they have liability insurance, while other states only need proof of insurance when a driver has an accident or gets tickets. Most states require it. I believe only 3-5 states do not require motorists to carry liability coverage, but the laws change annually in each state.
The simple answer is yes, most states require drivers to buy liability insurance. You have to purchase the minimum required by state law. Our article titled, How Much Car Insurance Should You Buy? shows a table with each state's minimum requirements.
In addition, if you visit this free car insurance quote page and select your state, you can see the Minimum Requirements.
Comment Update: Every state requires insurance or that you meet financial responsibility requirements. The only state without compulsory insurance laws is New Hampshire. Click on those states here to see the coverage options for each state. Each state changes/renews the laws annually, so some states that had no insurance requirements in the past now do. Here is a great link for car insurance mashup requirements.
Please check the Last Updated date below, as state laws change regularly.