I am claiming my child as a dependent, but the 1098T and 1099Q came in their name. Who should file these on their taxes? Parent or dependent?

The recipient files the 1099-Q. The parent reports the 1098-T to get education credit since the dependent, if claimed, is not eligible for the credit. 

If you claim your student as your Dependent, you can add their 1098-T and 1099-Q to your return.  However, you may not need to do so for one or both of these documents.

If the 1098-T has a larger amount in Box 1 (Tuition) than in Box 5 (Scholarships), you may qualify for an Education Credit, so you would want to add this to your return.  

If the funds from the 1099-Q were all used for Qualified Education Expenses (including Room & Board), neither your student or you need to enter it in TurboTax, just save it for your records.

Here's more info on How to Enter a 1098-T and Form 1099-Q. 

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