If I buy a house and let my adult children live there rent free or at rent?

Would have some amount of income yes.

Renting at an amount less than the fair market value, refer to Publication 527

Not Rented for Profit

If you do not rent your property to make a profit, you can deduct your rental expenses only up to the amount of your rental income. You cannot deduct a loss or carry forward to the next year any rental expenses that are more than your rental income for the year.

Where to report. Report your not-for-profit rental income on Form 1040 or 1040NR, line 21. For example, if you are filing Form 1040, you can include your mortgage interest and any qualified mortgage insurance premiums (if you use the property as your main home or second home), real estate taxes, and casualty losses on the appropriate lines of Schedule A (Form 1040) if you itemize your deductions.

www.irs.gov/publications/p527/ch04.html#en_US_2012_publink1000219164

Hope that you find the above enclosed information useful. 05/13/2013

If you take rent from them at below market rates, the full amount of rent you receive is income to you. You may take expenses only as miscellaneous deductions on Schedule A, subject to a 2% of AGI floor, and your deduction cannot be more than the amount of rent received. This type of rental is treated the same as hobby income.

If you provide your adult children with a gift of housing in excess of $14K per person per year, then you should file Form 709 Gift Tax return. No tax will be owed until your lifetime exclusion is used, which is over $5 million. If several persons live at the address, it may be that the value of the rent is less than $14K per person, so no gift tax issues arise.

In no case is there income tax liability to the recipients of the free or reduced rent.

renting to your relatives at lower than market, you claim the income but do not claim any expenses

You might. If the value is more than $14K for the year, you would file a gift tax form.

No. That is family and no law requires you to charge them rent.

If they pay rent, it's income to you.