Xbox One: Xbox One's games this month will be Massive Chalice and PoolnationFX (lol we've already have the 2nd one for 3 months, so boo). Both these games will be available all month
Xbox 360: Now the 360 will be getting some good days (it always does). It'll be getting Just Cause 2, and The Thief. Just Cause 2 will be available from June 1-15, while Thief will be available from June 16-30.