St. Peter, by Paolo Emilio Besenzi (1608-1656) [public domain / Wikimedia Commons]
A great deal of biblical data suggests a primacy and leadership role of St. Peter in the apostolic Church. The great Protestant scholar James D. G. Dunn stated, along these lines:
[I]t is Peter who becomes the focal point of unity for the whole Church – Peter who was probably the most prominent among Jesus’ disciples, . . . Peter who was the leading figure in the earliest days of the new sect in Jerusalem, . . . he became the most hopeful symbol of unity for that growing Christianity which more and more came to think of itself as the Church Catholic.
(Unity and Diversity in the New Testament, London: SCM Press, 2nd edition, 1990, 385-386)
I have collected 50 of these evidences in longer articles, but here, I shall present 20 of the best:
1. Matthew 16:18 (RSV) And I tell you, you are Peter, and on this rock I will build my church; and the powers of death shall not prevail against it.
The rock (Greek, petra) is St. Peter himself, not his faith. Jesus is the Architect who “builds.” Today, the overwhelming consensus of biblical commentators of all stripes favors this traditional Catholic understanding. St. Peter is the foundation-stone of the Church, making him head and superior of the family, but not founder of the Church; administrator, but not Lord of the Church.
2. Matthew 16:19 I will give you the keys of the kingdom of heaven . . .
The “power of the keys” (according to many Bible commentators) has to do with ecclesiastical discipline and administrative authority with regard to the requirements of the faith, as in Isaiah 22:22 (cf. Is 9:6; Job 12:14; Rev 3:7). This entails the use of excommunication, absolution, imposition of penances, and legislative powers. In the Old Testament a steward, or prime minister is a man who is “over a house” (Gen 41:40; 43:19; 44:4; 1 Ki 4:6; 16:9; 18:3; 2 Ki 10:5; 15:5; 18:18; Is 22:15, 20-21).
3. Matthew 16:19 . . . whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.
“Binding” and “loosing” were originally technical rabbinical terms, which meant to “forbid” and “permit” with reference to the interpretation of the law, and secondarily to “condemn” or “acquit.” Thus, St. Peter (and by logical extension, future popes) is given the authority to determine binding rules for the Church’s doctrine and life. “Binding and loosing” represent the legislative and judicial powers of the papacy and the bishops (Mt 18:17-18; Jn 20:23), and the power to absolve. St. Peter, however, is the only apostle who receives these powers by name and in a singular sense, making him preeminent.
4. Peter alone among the apostles receives a new, solemnly conferred name, Rock, (Jn 1:42; Mt 16:18).
5. St. Peter’s name occurs first in all lists of apostles (Mt 10:2; Mk 3:16; Lk 6:14; Acts 1:13). Matthew even calls him the “first” (10:2). Judas Iscariot is invariably mentioned last. This means something.
6. Christ teaches from Peter’s boat, and a miraculous catch of fish follows (Lk 5:1-11): perhaps a metaphor for the pope as a “fisher of men” (cf. Mt 4:19).
7. Peter was the first apostle to enter the empty tomb of the risen Jesus (Jn 20:6).
8. St. Peter is specified by an angel as the leader and representative of the apostles (Mk 16:7: “tell his disciples and Peter . . .”).
9. Peter is regarded by Jesus as the Chief Shepherd after Himself (Jn 21:15-17: “Feed my lambs . . . Tend my sheep . . . feed my sheep.”), singularly by name, and over the universal Church, even though others have a similar but subordinate role (e.g., Acts 20:28; 1 Pet 5:1-2).
10. Peter alone among the apostles is mentioned by name as having been prayed for by Jesus Christ in order that his “faith may not fail” (Lk 22:32).
11. Peter alone among the apostles is exhorted by Jesus to “strengthen” the Christian “brethren” (Lk 22:32).
12. St. Peter is the first to speak (and only one recorded) after Pentecost, so he was the first Christian to “preach the gospel” in the Church era (Acts 2:14-36).
13. Peter works the first miracle of the Church Age, healing a lame man (Acts 3:6-12).
14. Peter is regarded by the common people as the leader of Christianity (Acts 5:15: “as Peter came by at least his shadow might fall on some of them.”).
15. Peter was the first traveling missionary, and first to exercise the “visitation of the churches” (Acts 9:32-38, 43). Paul’s missionary journeys begin in Acts 13:2.
16. Cornelius is told by an angel to seek out St. Peter for instruction in Christianity:
Acts 10:21-22 And Peter went down to the men and said, “I am the one you are looking for; what is the reason for your coming?”  And they said, “Cornelius, a centurion, an upright and God-fearing man, who is well spoken of by the whole Jewish nation, was directed by a holy angel to send for you to come to his house, and to hear what you have to say.”
17. Peter is the first to receive the Gentiles into the fellowship of the Christian Catholic Church, after a revelation from God (Acts 10:9-48).
18. Peter presides over and is preeminent in the first Church-wide council of Christianity (Acts 15:7-11).
19. Paul distinguishes the Lord’s post-Resurrection appearances to St. Peter from those to other apostles (1 Cor 15:4-8). The two disciples on the road to Emmaus do the same (Lk 24:34), even though they themselves had just seen the risen Jesus within the previous hour (Lk 24:33).
20. St. Peter’s name is mentioned more often than all the other disciples put together: 191 times (162 as Peter or Simon Peter, 23 as Simon, and 6 as Cephas). John is next in frequency with only 48 appearances.