I have lived in the same place for 3years and now the owner is selling the property. What happens if the property is bought and I move, does the new owners pay me my deposit back that I paid to the old owner, or how does that work?

First of course if you have a written lease the new owners will be obligated to honor the lease.

Also if you are on a month to month rental, it is possible the current owner may merely give you notice to be out by a certain date which will be close to the closing date.

If not at closing the deposit on your rental is suppose to transfer to the buyer ( it is normally done merely as a credit on the closing statement, not actual money passing hands)

But it is normal that you will get your deposit back from the new owner if you move after they own the property.