The earliest known Papal coinage began under Pope Saint Gregory III (731-741 AD), however, the first serious Papal coinage began under Pope Adrian I (772-795) at the mint in Rome and continued through Pope Benedict VII (974-983). From the reign of Pope John XIV (983-984) to Pope Clement IV (1265-1268), no known papal issues exist. This is due to the Roman Senate taking over control of the mint output and issuing anonymous coinage. Papal coinage was restored in 1268, but it was a period of Sede Vacante (no pope) until 1271 and coins were issued as such. Even after Pope Gregory X was named in 1271, no Papal coinage is known again until Pope Boniface VIII (1296-1303) and continues yet today.
Along with the fascinating and artistic Papal coins, there are also the coins of the intriguing antipopes. Antipopes were religious leaders who claimed to be the true leader of the Church, against the named pope in Rome. Although it didn’t happen often, there were times and periods where the position of the Church in Rome was in opposition to other widely-held views to such an extent that a large group would instead recognize a different leader. Antipope St. Hippolytus (217-235) is considered the first antipope. The first antipope of which coinage is “known” is Antipope Christopher (903-904), however, no authentic examples are yet known. The first antipope to issue serious coinage was Antipope Clement VII (1378-1394) – the first antipope of the Great Schism (1378-1417).
Lastly there was a short-lived Second Roman Republic in 1849, which was fostered by the French Revolution, and issued coins from the mint in Rome.
If you are interested in Papal coinage, the must-have reference is “Papal Coins” by Allen G. Berman.