The Catholic Church of Iraq has no national (Latin) episcopal conference, but is united in an inter-rite Assembly of the Catholic Bishops of Iraq, given its diversity :

• a Latin non-Metropolitan Archdiocese (participation in the transcontinental Arab Region Latin Bishops conference)
• divided over four Eastern Catholic rite-specific particular churches : a Patriarchate, two Metropolitan - and six other archeparchies, three more eparchies and two (pre-diocesan) Patriarchal exarchates.

There is an Apostolic Nunciature to Iraq in the national capital Baghdad, as papal diplomatic representation at embassy-level (Established as Apostolic Delegation of Mesopotamia, Kurdistan and Lesser Armenia, in 1937 renamed as Apostolic Delegation of Iraq, promoted on 1966.10.14), into which is also vested the Apostolic Nunciature to neighbouring (Trans)Jordan.

• Archdiocese of Baghdad (non-Metropolitan)

(Chaldean - = Syro-Oriental Rite)

• Chaldean Catholic Patriarchate of Babylon (at Al-Mansour, Baghdad; initially Diocese of Seleucia-Ctesifonte)

• Metropolitan Archeparchy of Baghdad
  • Chaldean Catholic Eparchy of Alquoch
  • Chaldean Catholic Eparchy of Amadiyah and Zaku
  • Chaldean Catholic Eparchy of Aqra

• nominally Metropolitan Chaldean Catholic Archeparchy of Kirkuk-Sulaimaniya (no actual suffragan)

• Chaldean Catholic Archeparchy of Basra (Bassorah, successor of Perat-Maishan )
• Chaldean Catholic Archeparchy of Arbil
• Chaldean Catholic Archeparchy of Mosul

(no Metropolitan, no Suffragan)

Syrian Catholic (Antiochian Rite)

• Syrian Catholic Archeparchy of Baghdad
• Syrian Catholic Archeparchy of Mosul
• Patriarchal Exarchate of Basra and the Gulf

Armenian Catholic (Armenian rite)

• Archeparchy of Baghdad

(Greek-)Melkite Catholic (Byzantine Rite)

• Melkite Catholic Patriarchal Exarchate of Iraq

• Two Episcopal Titular bishoprics : Anbar of the Chaldeans, Hirta (now Latin)

TO BE COMPLETED?

Apart from the precursors of the current sees :

Latin :

• Mission sui iuris of Mossul (suppressed)
• Apostolic Prefecture of Syria and Cilicia (formerly and later Mission sui iuris)

Eastern Catholic :

• Chaldean Catholic Eparchy of Sulaimaniya (merged into Kirkuk)
• Chaldean Catholic Eparchy of Zaku (Zākhō; merged into Amadiyah = Amadia)

• List of Catholic dioceses (structured view)
• Christianity in Iraq

• GCatholic.org - data for all sections.
• Catholic-Hierarchy entry.