According to Catholic tradition, the legitimacy and supremacy of the Pope over all other bishops is founded upon the following Biblical passage, from Matthew 16:18-19…

18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.

19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

Any subsequent loss of these keys notwithstanding, these two simple lines have supposedly justified an enormously important aspect of Western (and world) history. But their authenticity is anything but certain. Barbara Walker writes:

It is now certain that there was no St. Peter in Rome to “found the papacy.” Stories about Peter were invented after the Roman See was well established. During the first five centuries of the Christian era, no one thought the bishop of Rome had a right to govern other bishops; there was no such doctrine as the primacy of the Roman See. “Christ neither founded nor desired the Church.” (p. 791)

Richard McBrien, a Catholic historian and theologian at the Universeity of Notre Dame, wrote Lives of the Popes which sings a similar song:

Although Catholic tradition, beginning in the late second and early third centuries, regards St. Peter as the first Bishop of Rome and, therefore, as the first pope, there is no evidence that Peter was involved in the initial establishment of the Christian community in Rome (indeed, what evidence there is would seem to point to the opposite direction) or that he served as Rome’s first bishop. Not until the pontificate of St. Pius I in the middle of the second century (ca. 142-ca. 155) did the Roman church have a monoepiscopal structure of government (one bishop as pastoral leader of a diocese).Those whom Catholic tradition lists as Peter’s immediate successors (Linus, clement, et al.) did not function as the one bishop of Rome. […] The Roman community seems instead to have had a corporate or collegial form of pastoral leadership. […] In any case, the popes of the first four centuries—that is, until the watershed papacy of Leo I in the middle of the fifth century—functioned with relatively limited authority beyond Rome and its immediate environs. (page 25)

More quotes from that source can be found here. This is also an interesting quote:

Neither is there any evidence that the bishops of Rome actually governed other local churches, legislated for them, or appointed their bishops. At most, the bishops of Rome during those first four centuries may have exercised a kind of metropolitan authority over neighboring Italian sees, which came to be known as suburbicarian sees. But there is less evidence even for this that there is for the clearly “sovereign” authority exercised by the see of Alexandria over all the churches of Egypt and Cyrenaica. (page 2)

There’s an interesting theory which suggests that the so-called Petrine passage from Matthew 16 above was actually itself a later forgery. It seems to only appear in Matthew of all the four Gospels. The argument is a little bit obtuse, but so are pretty much all Biblical arguments. It seems that in the Gospel According to John 1:42, Jesus first meets Peter. At the time though, his name is Simon, and Jesus renames him “Cephas” which is Aramaic for “stone”. At this point, Jesus does not also tell Peter he’s going to found a Church on him. Matthew’s anecdote is supposed to happen something like a year or more after this incident. Further, there’s some kind of argument about the name Peter being a Greek pun on the word “rock” rather than Aramaic.

I don’t completely follow the subtleties of that argument, but if you want to look more into it give it a shot. I am, however, pretty interested in the whole argument that the Catholic Church was originally not quite so rigidly hierarchical as it is today. Nor was it’s power center the city of Rome:

In the first three centuries of the Christian era, Rome was not an especially important centre for the Faith. The great sees of the early Christian world were Alexandria, Ephesus and Antioch – each a centre of early proselytising and of a large Christian community. Each claimed some justifying link to an apostle. Paul had lived in Antioch, for example and John, it was said, had seen out his days in Ephesus. Mark was associated with Alexandria. Rome, in contrast, had no direct apostolic connection.

It’s also pretty interesting to look at the Roman tradition of syncretically fusing religions together. It seems Christianity was just the newest gambit in this centuries-old game of changing with the tides:

Remnants and residue of the old religions became part of the pope’s inheritance, the fixtures and fittings of the previous tenant. Images of Peter were fashioned after traditional statues of Jupiter. The mystery religions had had symbolic keys to the inner mysteries; keys now found their way into papal regalia. The pontiffs inherited a throne, decorated with the twelve labours of Hercules, which the Popes used for centuries. The Pope’s retinue of cardinals took on the title from ‘cardo’, meaning hinge, from the hinge on the door guarded by Janus, the god of entrances. Peter himself, of course, would similarly become the doorman on the ‘pearly gates’ of heaven.

I’ve written more along this line elsewhere for anybody interested. The other direction worth following in all of this would of course be the Protestant Reformation. During that time, all kinds of new arguments against the Papacy and the Petrine Supremacy were researched and explicated. Many Protestant churches reverted back to the governing structures of the early Catholic Church befor the primacy of the Roman See was really established. Episcopalianism and Presbyterianism are great examples, as are many other denominations. My all time favorite Protestant doctrine though is anti-sacerdotalism, or “the priesthood of all believers,” which is a quasi-gnostic doctrine that says you don’t need priests to intercede on your behalf. In other words: Jesus didn’t just give Peter the keys to Heaven - he gave them to all of us. Or, even better, he taught us how to use the keys that were already inside us to unlock the Kingdom which lay all around.