The State of Illinois issues a special driver's license to non-citizens who cannot get a SSN. The license is called Temporary Visitor Driver's License (TVDL). You can apply for a TVDL if you have a lawful immigration status. If you are an undocumented immigrant, you may be able to get a TVDL as well. A TVDL cannot serve as a government-issued ID.
TVDL is valid for three years. After that, you have to re-apply as a new applicant.
