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The Pope is the Bishop of Rome and the leader of the worldwide Catholic Church. He is the head of state of the Vatican. The exorcists are considered the personal army of the pope.
When the Third Exorcist Program successfully produced half-Akuma species (Madarao, Tewaku, Tokusa, Kiredori and Goushi being the first results), Renny Epstain told Komui that the Pope himself ordered to put these five in service as the new Exorcists of the Black Order.
When Allen Walker and the Noah escape the cell of the Order, the Pope adressed his command about the situation to the whole Order (namely, about cancelling Allen's rights as an exorcict and classifying him as a Noah). In the anime, the message from the pope is seen being given to the Lead Cardinal. 
- The pope didn't appear in the manga yet. Only his decisions are reported by other characters without official confirmation that they really come from him.