I want to gather together a collection of links to some posts I've written on the papacy over the years.
Here's an overview of the Biblical evidence pertaining to the papacy. And here's a post addressing the earliest extrabiblical sources.
I wrote a series on apostolic succession, and that series discusses many subjects relevant to the papacy. There's a lot of material on how Hegesippus, Irenaeus, and other early sources viewed Peter, the nature of succession, and other pertinent issues. And here's a post on the episcopate in the early church.
Here's something I wrote about how Peter can be viewed, and often has been viewed, as having successors in a variety of ways. Many patristic and later sources have referred to successors of Peter who have no papal authority. You can believe in a notion of Petrine succession without believing in a papacy.
Catholics often argue for a papacy by claiming that Peter is always listed first in the New Testament's lists of apostles. I've written a response.
I also wrote a post on Isaiah 22 and the papacy.
Here's something I wrote about how we could argue for a Pauline papacy and Ephesian primacy by using arguments like those Catholics use to argue for a Petrine papacy and Roman primacy. And here's something else I wrote about how Biblical and patristic passages about somebody other than Peter or a Roman bishop could be interpreted in a papal sense if we applied Catholic reasoning consistently.
I've often addressed whether individual church fathers believed in the concept of the papacy. Here and here are a couple on Clement of Rome. And Hegesippus. On Irenaeus, see the comments section of the thread here and the post here. Regarding Origen, see this and this. Cyprian. On Augustine, see here, here, and here.
Another way to approach the patristic evidence is by asking how the Christians of the patristic era viewed issues of unity. Do they seem to have been living under a papal system of church government? See here and here.
It's sometimes claimed that all of the earliest Roman bishops died as martyrs. I addressed that subject in the post here.
I also wrote about conciliarism and rejection of the papacy in the medieval era.
Catholics often appeal to the notion of development of doctrine to explain the lack of early evidence for a papacy. Here's a post I wrote on the subject. I also wrote a five-part series on comparisons between the development of the canon of scripture and the development of a Catholic view of the church, including the papacy: one, two, three, four, five.
Finally, here's a post that gives some examples of how corrupt many Popes have been over the centuries and how they've often failed to meet Biblical and patristic standards for being a bishop.