What is the Biblical basis for Peter being the leader of the apostles?

The Roman Catholic church claims that the apostle Peter was the first bishop of Rome, that he was the leader of the other apostles/bishops, and that since the pope is the rightful heir of Peter's title (according to the Church), the pope would also be the leader of the bishops. What is the biblical basis for this claim? Does the Church take any specific verse as evidence for this claim?