If you have an existing Illinois tax liability, make full payments between October 1, 2019, and November 15, 2019 and your penalties and interest will be waived.

Eligible liabilities are taxes due from periods ending after June 30, 2011, and prior to July 1, 2018. If you failed to file a tax return or incorrectly reported the liability due on a previously filed return for these tax periods, now is the time to file returns, make corrections, and pay the tax. Eligible penalties and interest will be waived.

You must file an original return for non-filed periods or file an amended return to make corrections.

The Illinois Department of Revenue has a Bulletin on the topic here, and a related FAQ here.