Although some tickets and actions can't be removed, most can once the conviction is at least three years old. The period is longer for folks with a commercial driver's license. If the ticket resulted in a suspension or revocation of your driving privilege then five years must pass from the suspension or revocation reinstatement date before the ticket(s) can be purged.
Tickets for Failure to Maintain Insurance cannot be removed from a driver record. DWI cases are handled through a different process.
Most people take care of this themselves. You can request to have old tickets purged from your record using one of the following methods: