In the Byzantine Empire, an exarch was a proconsul or viceroy who governed a province at some remove from the central authorities, the Emperor and the Patriarch of Constantinople. In the ecclesiastical organization of the Empire of the East, the exarch of the political division called a "diocese" was in the 4th and 5th centuries the same as a "primate," a dignity that was intermediate between a patriarch and the metropolitan bishops, the term "patriarch" being formally restricted after 451 CE to the chief bishops of a few most important cities.
The Byzantine Exarchs of Africa nominally governed Sardinia and Corsica.
The Exarchs of Ravenna
Ravenna had become the capital of the western Roman Empire in 404 under Honorius. It remained the capital of Italy under the Ostrogoths, and after the reconquest became the seat of the provincial governor (539). Ravenna remained the seat of the Exarch until the revolt of 727 over Iconoclasm. Thereafter, the growing menace of the Lombards and the split between eastern and western Christendom that Iconoclasm caused made the position of the Exarch more and more untenable. The last Exarch was killed by the Lombards in 751. See Exarchate of Ravenna.
In the Orthodox Church, an exarch is still a prelate: an inspector of monasteries , a deputy of the Patriarch or in many cases he rules a foreign Church on behalf of a Patriarchate, e.g. the Serbians, Romanians, Bulgarians, the Jerusalem Patriarchate et. al , all have exarchates in the USA. The style of the exarchs of the Patriarchate of Jerusalem is " Exarch of the Holy Sepulcher"
The Exarchate of Ravenna was not the sole Byzantine province in Italy.
His ally Pippin the Younger, King of the Franks, donated the conquered lands of the former exarchate to the Papacy in 756; this donation, which was confirmed by his son Charlemagne in 774, marked the beginning of the temporal power of the popes as the Patrimony of Saint Peter.
So the Exarchate disappeared, and the small remnants of the imperial possessions on the mainland, Naples and Calabria, passed under the authority of the Catapan of Italy, and when Sicily was conquered by the Arabs in the 9th century the remnants were erected into the themes of Calabria and Langobardia.