'They are seeking refuge in the relatively safe Kurdish-controlled region in the north.
The UN High Commission for Refugees said about 1,000 families have left Baghdad and Mosul province since an attack on a church left 68 people dead.
It said the flight of Christians to other parts of Iraq and abroad has become "a slow but steady exodus".
The UNHCR also said it was dismayed that European governments are deporting failed Iraqi asylum seekers to areas of the country it does not consider safe.'
This is very sad, and so close to Christmas.