The Catholic Church, in her liturgy, lists the early bishops of Rome after Peter: Linus, Cletus, Clement.
Regarding Clement, Methodist scholar Ben Witherington (Asbury Seminary) writes:
Clement of Rome is an interesting figure in early Christian history, not least because he had contact with some of the original Christian eyewitnesses, apostles and their co-workers. Hermas mentions this Clement (Shepherd 8.2) as a leader in the church in Rome to whom he sent a book, who was then to “distribute it to churches in other locations, for that is his commission”. Tertullian in fact says he was the second (or third?) bishop/overseer of the church in Rome ordained by no less than Peter himself (Prescription 32). Origen (Comm. John 6.36) says he was the sometime companion of Paul which comports with Phil. 4.3 which suggests Clement is with Paul in Rome. Indeed, what Phil. 4.3 says is that Clement is a co-worker of Paul in the Gospel. It is thus important that we take with absolute seriousness what Clement says about the leadership structure of early Christianity from the time it began right on through the first century. Most scholars date 1 Clement to the last decade of the first century, though it may be from the 80s….
[continue reading about the Leadership Structure of Earliest Christianity]