The early churches did talk about apostolic succession, but not the way the Roman Catholics do today.
The Roman Catholics argue that it is the passing down of authority. Peter, and not any other apostles, passed authority down to the first bishop of Rome, and not any other bishops, and then down to the succeeding bishops of the Roman church.
This is not what the early church believed.
To the early churches, apostolic succession was a proof of the preservation of truth within the churches.
Peter, for example, taught the truth—the faith once for all delivered to the saints—to Linus and other elders in Rome around A.D. 60. Linus taught it to Anacletus, Anacletus taught it to Clement, and so forth.
And this was just as true in every other church started by an apostle:
In the late 2nd and early 3rd century, apostolic succession was a great argument.
After all, who really understands the Scriptures and the message of the apostles? Is it not those who are directly descended—spiritually speaking—from those to whom the apostles committed their message and the churches themselves?
However, to argue that apostolic succession has faithfully and accurately preserved apostolic tradition for two thousand years, including throughout the massively corrupt Middle Ages is quite a different issue.
The following four part series covers the early church's view of apostolic succession in depth. I think you'll enjoy it.
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