We have 1099-DIV statements for our children but we are the custodians of the accounts. Do I claim these on my tax return?

[AmyT]

You do not include your children's income on your tax return.

You can walk through this tool: https://www.irs.gov/uac/do-i-need-to-file-a-tax-return to determine if they are required to file.  (They most likely do not have to file returns.)

Even if they do need to file, you are still able to claim them as dependents on your return. It will be very important for you to answer the dependent questions on their returns as "Yes":

1. Someone else can claim me as a dependent on their tax return.
2. And this person will claim me on their 2016 tax return.