Yes, normally your car insurance will cover you if you move to a different city; however you need to notify your auto insurance company of the change of address.
Your car insurance provider will need your new address in your new city for a variety of reasons. One is for billing purposes; you don't want to miss your car insurance notifications, statements and payment notices. Also, your rates may be affected due to your move. If your zip code and geographical location are rating factors, which with most auto insurance companies it is, then moving to a new city within your same state may make your rates go up or it could make your rates go down.
How urban or rural your new city is could affect your rates as well as if you drive more or less now in your new city. For instance if you moved closer to your place of work or school and so drive less for your daily commute it may help your rates go down conversely if you moved farther away from your workplace or school and have a longer commute, then your car insurance rates may go up.
Car insurance rates are different for different neighborhoods and can have an impact on your finances if the rates are much higher in the area you are thinking of moving to then the area in which are moving from. Areas with high local taxes to support law enforcement and schools often also have lower crime rates and are therefore assigned lower risks than other neighborhoods. It is wise to check out what your auto insurance rates will be before you move so you can plan accordingly with your finances or even see if another part of town (with a different zip code) is cheaper than another. (See "How your ZIP code drives up your car insurance")
It's also important to tell your insurance provider about your move because if you fail to inform your car insurance company about your move and you are in an accident they may be able to deny claims due to misrepresentation about where you are now living and your vehicle is garaged.
Whether you are moving across town, across your state or across the country, when you move you should notify your car insurance company and also take the time to shop around and make sure you are getting the best car insurance rates possible.
CarInsurance.com can help you when comparing insurance rates. We offer multiple company insurance rates for personal auto insurance and the ability to purchase from one place; we are ahead of other online insurance companies. Start your quote to see how companies compete for your business so that you can obtain affordable auto insurance.