They call your credit card issuer and, pretending to be you, ask to change the mailing address on your credit card account.  The imposter then runs up charges on your account.  Because your bills are being sent to the new address, it may take some time before you realize there's a problem.

The open a new credit card account, using your name, date of birth and SSN.   When they use the credit card and don't pay the bills, the delinquest account is reported on your credit report.

The establish phone or wireless service in your name.

They open a bank account in your name and write bad checks on that account.

They file for bankruptcy under your name to avoid paying debts they've incurred under your name, or to avoid eviction.

They counterfeit checks or debit cards, and drain your bank account.

They buy cars by taking out auto loans in your name.

They give your name to the police during an arrest.  If they're released from police custody, but don't show up for their court date, an arrest warrant is issued in your name.