I am working on filing taxes abroad (I've moved to Europe). My daughter was born here and does not yet have an SSN. Does she need one in order for me to file taxes?

Yes, in order to claim her as a dependent, she would need a SSN.  

If you are married, your only filing options are Married Filing Jointly or Married Filing Separately. 

If they do not have an SSN or ITIN and you choose to file Married Filing Separate, you can try typing NRA in as your spouses last name.  Make sure you put a check mark in the box that says My spouse is a Non-Resident alien in the My info section. When you get to the end and try to transmit the return, ignore the e-file errors regarding your NRA spouse and select transmit your return. If that does not work you will need to print and mail your return and write NRA (Non Resident Alien) in the space for her SSN. 

If you file as Married Filing Separate, then you would only include your income. 

 If your spouse chooses to be treated as an US Citizen for tax purposes, and you want to file as Married Filing Joint, you would need to include their income on your return as well and they would be taxed on it. If you choose filing jointly, you will need to print and mail your return along with the W-7 and any other required documents to

IRS

ITIN Operation

P.O. Box 149342

Austin, TX 78714-9342

Claiming Non Citizen Spouse and Children on Your Taxes

How to obtain an ITIN