Ok, so to be specific - If you are wanting to change your name due to the marriage, on the I-485, Part 1 - Question 1 "Your Current Legal Name", put your new, married name here. (As long as the name is some combination of names shown on the marriage certificate) Question 2, "Other Names You Have Used Since Birth" put your maiden name. Then in Question 19, you check yes in all the boxes and fill in your SSN in the spot provided and when approved, USCIS will notify SSA of your name change, and they will issue you a new card in your married name. If you are submitting the I-765 for Employment Authorization, follow the same procedure on that form as well and you will receive your new SSN in your married name somewhat quicker as the I-765 is usually adjudicated faster than the I-485. The SSN you receive after the I-765 will still have the restriction "Valid for Employment Only With DHS Authorization" on it but that's ok because you'll have your Employment Authorization Document. You'll get the unrestricted SSN card after your AOS approval.