"The Church of God which sojourns in Rome to the Church of God which sojourns in Corinth....If anyone disobey the things which have been said by Him through us, let them know that they will involve themselves in transgression and in no small danger." Pope Clement of Rome [regn. c A.D.91-101], 1st Epistle to the Corinthians, 1,59:1 (c. A.D. 96).
"Thereupon Victor, who presided over the church at Rome, immediately attempted to cut off from the common unity the parishes of all Asia, with the churches that agreed with them, as heterodox; and he wrote letters and declared all the brethren there wholly excommunicate..." Pope Victor I [regn. A.D. 189-198], in Eusebius EH, 24:9 (A.D. 192).
"Stephen, that he who so boasts of the place of his episcopate, and contends that he holds the succession from Peter, on whom the foundations of the Church were laid...Stephen, who announces that he holds by succession the throne of Peter." Pope Stephen I [regn. A.D. 254-257], Firmilian to Cyprian, Epistle 74/75:17 (A.D. 256).
"I beseech you, readily bear with me: what I write is for the common good. For what we have received from the blessed Apostle Peter, that I signify to you; and I should not have written this, as deeming that these things were manifest unto all men, had not these proceedings so disturbed us." Pope Julius [regn. A.D. 337-352], To the Eusebians, fragment in Athanasius' Against the Arians, 2:35 (c. A.D. 345).
"Why then do you again ask me for the condemnation of Timotheus? Here, by the judgment of the apostolic see, in the presence of Peter, bishop of Alexandria, he was condemned, together with his teacher, Apollinarius, who will also in the day of judgment undergo due punishment and torment. But if he succeeds in persuading some less stable men, as though having some hope, after by his confession changing the true hope which is in Christ, with him shall likewise perish whoever of set purpose withstands the order of the Church. May God keep you sound, most honoured sons." Pope Damasus [regn. A.D. 366-384], To the Eastern Bishops, fragment in Theodoret's EH, 5:10 (c. A.D. 372).
"We bear the burdens of all who are heavy laden; nay, rather, the blessed apostle Peter bears them in us and protects and watches over us, his heirs, as we trust, in all the care of his ministry....Now let all your priests observe the rule here given, unless they wish to be plucked from the solid, apostolic rock upon which Christ built the universal Church....I think, dearest brother, disposed of all the questions which were contained in your letter of inquiry and have, I believe, returned adequate answers to each of the cases you reported by our son, the priest Basianus, to the Roman Church as to the head of your body....And whereas no priest of the Lord is free to be ignorant of the statutes of the Apostolic See and the venerable provisions of the canons." Pope Sircius [regn. c A.D. 384-399], To Himerius, bishop of Tarragona (Spain), 1,3,20 (c. A.D. 392).
"Care shall not be lacking on my part to guard the faith of the Gospel as regards my peoples, and to visit by letter, as far as I am able, the parts of my body throughout the divers regions of the earth." Pope Anastasius [regn. A.D. 399-401], Epistle 1 (c. A.D. 400).
"In making inquiry with respect to those things that should be treated ... by bishops ... as you have done, the example of ancient tradition ... For you decided that it was proper to refer to our judgment, knowing what is due to the Apostolic See, since all we who are set in this place, desire to follow that Apostle from whom the very episcopate and whole authority of this named derived ... that whatsoever is done, even though it be in distant provinces, should not be ended without being brought to the knowledge of this See, that by its authority the whole just pronouncement should be strengthened, and that from it all other Churches (like waters flowing from their natal source and flowing through the different regions of the world, the pure streams of one incorrupt head)...you also show your solicitude for the well being of all, and that you ask for a decree that shall profit all the Churches of the world at once." Pope Innocent I [regn. A.D. 401-417], To the Council of Carthage, 1,2 (A.D. 417).
"It is therefore with due care and propriety that you consult the secrets of the Apostolic office that office, I mean, to which belongs, besides the things which are without, the care of all the Churches...Especially as often as a question of faith is discussed, I think that all our brothers and fellow bishops should refer to none other than to Peter, the author of their name and office." Pope Innocent I [regn. A.D. 401-417], To the Council of Mileve, 2 (A.D. 417).
"Although the tradition of the fathers has attributed to the Apostolic See so great authority that none would dare to contest its judgment, and has preserved this ever in its canons and rules, and current ecclesiastical discipline in its laws still pays the reverence which it ought to the name of Peter...For he himself has care over all the churches, and above all of that which he sat...Since, then Peter is the head of so great authority, and has confirmed the suffrages of our forefathers since his time...and as bishops you are bound to know it; yet; though such was our authority that none could reconsider our decision." Pope Zosimus [regn. A.D. 417-418], To the Council of Carthage (c. A.D. 418).
"For it has never been lawful to reconsider what has once been settled by the apostolic see." Pope Boniface [regn. A.D. 418-422], To Rufus bishop of Thessalonica (c. A.D. 420).
"The universal ordering of the Church at its birth took its origin from the office of blessed Peter, in which is found both directing power and its supreme authority. From him as from a source, at the time when our religion was in the stage of growth, all churches received their common order. This much is shown by the injunctions of the council of Nicea, since it did not venture to make a decree in his regard, recognizing that nothing could be added to his dignity: in fact it knew that all had been assigned to him by the word of the Lord. So it is clear that this church is to all churches throughout the world as the head is to the members, and that whoever separates himself from it becomes an exile from the Christian religion, since he ceases to belong to its fellowship." Pope Boniface [regn. A.D. 418-422], To the bishops of Thessaly (c. A.D. 420).
"None has ever been so rash as to oppose the apostolic primacy, the judgment of which may not be revised; none rebels against it, unless he would judge in his turn." Pope Boniface [regn A.D. 418-422], To Rufus and bishops of Macedonia (c. A.D. 420).
"Wherefore, assuming to yourself the authority of our see and using our stead and place with power, you will deliver this sentence with utmost severity." Pope Celestine [regn A.D. 422-427], To Cyril of Alexandria, Epistle 1 1 (A.D. 430).
"The blessed apostle Peter, in his successors, has handed down what he received. Who would be willing to separate himself from the doctrine of whom the Master himself instructed first among the apostles?" Pope Sixtus III, [regn A.D. 432-440], To John of Antioch (A.D. 433).
"But this mysterious function the Lord wished to be indeed the concern of all the apostles, but in such a way that He has placed the principal charge on the blessed Peter, chief of all the Apostles: and from him as from the Head wishes His gifts to flow to all the body: so that any one who dares to secede from Peter's solid rock may understand that he has no part or lot in the divine mystery." Pope Leo the Great [regn. A.D.440-461], Epistle 10 (A.D 445).
"And so he too rejoices over your good feeling and welcomes your respect for the Lord's own institution as shown towards the partners of His honour, commending the well ordered love of the whole Church, which ever finds Peter in Peter's See, and from affection for so great a shepherd grows not lukewarm even over so inferior a successor as myself." Pope Leo the Great [regn. A.D.440-461], Sermon 2 (A.D ante 461).
"'Thou art the Christ, the Son of the living God,' and every tongue which confesses the Lord, accepts the instruction his voice conveys. This Faith conquers the devil, and breaks the bonds of his prisoners. It uproots us from this earth and plants us in heaven, and the gates of Hades cannot prevail against it. For with such solidity is it endued by God that the depravity of heretics cannot mar it nor the unbelief of the heathen overcome it." Pope Leo the Great [regn. A.D.440-461], Sermon 3:2-3 (A.D ante 461).
"Who does not cease to preside in his see, who will doubt that he rules in every part of the world." Pope Leo the Great [regn. A.D.440-461], Sermon 5 (A.D ante 461).
And who can forget what the Council of Chalcedon stated explicitly:
"Peter speaks through Leo!" (Chalcedon, 451 AD).
Also on the ScriptureCatholic Website we find these quotes as well:
"Through envy and jealousy, the greatest and most righteous pillars [of the Church] have been persecuted and put to death. Let us set before our eyes the illustrious apostles. Peter, through unrighteous envy, endured not one or two, but numerous labours and when he had at length suffered martyrdom, departed to the place of glory due to him."Clement of Rome, The First Epistle of Clement, 5 (c. A.D. 96).
"I do not, as Peter and Paul, issue commandments unto you." Ignatius of Antioch, Epistle to the Romans, 4 (c. A.D. 110).
'You have thus by such an admonition bound together the plantings of Peter and Paul at Rome and Corinth." Dionysius of Corinth, Epistle to Pope Soter, fragment in Eusebius' Church History, II:25 (c. A.D. 178).
"Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter and Paul were preaching at Rome, and laying the foundations of the Church." Irenaeus, Against Heresies, 3:1:1 (c. A.D. 180).
"As Peter had preached the Word publicly at Rome, and declared the Gospel by the Spirit, many who were present requested that Mark, who had followed him for a long time and remembered his sayings, should write them out." Clement of Alexandria, fragment in Eusebius Church History, VI:14,6 (A.D. 190)
"It is, therefore, recorded that Paul was beheaded in Rome itself, and that Peter likewise was crucified under Nero. This account of Peter and Paul is substantiated by the fact that their names are preserved in the cemeteries of that place even to the present day. It is confirmed likewise by Caius, a member of the Church, who arose under Zephyrinus, bishop of Rome. He, in a published disputation with Proclus, the leader of the Phrygian heresy, speaks as follows concerning the places where the sacred corpses of the aforesaid apostles are laid: 'But I can show the trophies of the apostles. For if you will go to the Vatican or to the Ostian way, you will find the trophies of those who laid the foundations of this church.'" Gaius, fragment in Eusebius' Church History, 2:25 (A.D. 198).
"[W]hat utterance also the Romans give, so very near (to the apostles), to whom Peter and Paul conjointly bequeathed the gospel even sealed with their own blood." Tertullian, Against Marcion, 4:5 (inter A.D. 207-212).
'We read the lives of the Caesars: At Rome Nero was the first who stained with blood the rising blood. Then is Peter girt by another (an allusion to John 21:18), when he is made fast to the cross." Tertullian, Scorpiace, 15:3 (A.D. 212).
"Peter...at last, having come to Rome, he was crucified head-downwards; for he had requested that he might suffer this way." Origen, Third Commentary on Genesis, (A.D. 232).
"Thus Peter, the first of the Apostles, having been often apprehended, and thrown into prison, and treated with igominy, was last of all crucified at Rome." Peter of Alexandria, The Canonical Epistle, Canon 9 (A.D. 306).
"[W]hich Peter and Paul preached at Rome..." Lactantius, The Divine Institutes, 4:21 (A.D. 310).
"Peter...coming to the city of Rome, by the mighty cooperation of that power which was lying in wait there..." Eusebius, Ecclesiastical History, II:14,5 (A.D. 325).
"This man [Simon Magus], after he had been cast out by the Apostles, came to Rome...Peter and Paul, a noble pair, chief rulers of the Church, arrived and set the error right...For Peter was there, who carrieth the keys of heaven..." Cyril of Jerusalem, Catechetical Lectures ,6:14-15 (c. A.D. 350).
"And Peter, who had hid himself for fear of the Jews, and the Apostle Paul who was let down in a basket, and fled, when they were told, 'Ye must bear witness at Rome,' deferred not the journey; yea, rather, they departed rejoicing..." Athanasius, Defence of his Flight, 18 (c. A.D. 357).
"I think it my duty to consult the chair of Peter, and to turn to a church whose faith has been praised by Paul...My words are spoken to the successor of the fisherman, to the disciple of the cross." Jerome, To Pope Damasus, Epistle 15 (A.D. 377).
“For if when here he loved men so, that when he [Peter] had the choice of departing and being with Christ, he chose to be here, much more will he there display a warmer affection. I love Rome even for this, although indeed one has other grounds for praising it, both for its greatness, and its antiquity, and its beauty, and its populousness, and for its power, and its wealth, and for its successes in war. But I let all this pass, and esteem it blessed on this account, that both in his lifetime he wrote to them, and loved them so, and talked with them whiles he was with us, and brought his life to a close there.” John Chrysostom, Epistle to the Romans, Homily 32 (c. A.D. 391).
"Which was mere to the interest of the Church at Rome, that it should at its commencement be presided over by some high-born and pompous senator, or by the fisherman Peter, who had none of this world's advantages to attract men to him?" Gregory of Nyssa, To the Church at Nicodemia, Epistle 13 (ante A.D. 394).
"But some people in some countries of the West, and especially in the city, [Rome] not knowing the reason of this indulgence, think that a dispensation from fasting ought certainly not to be allowed On the Sabbath, because they say that on this day the Apostle Peter fasted before his encounter with Simon [Magus]." John Cassian, Institutes, X (ante A.D. 435).
"The whole world, dearly-beloved, does indeed take part in all holy anniversaries [of Peter & Paul], and loyalty to the one Faith demands that whatever is recorded as done for all men's salvation should be everywhere celebrated with common rejoicings. But, besides that reverence which to-day's festival has gained from all the world, it is to be honoured with special and peculiar exultation in our city, that there may be a predominance of gladness on the day of their martyrdom in the place where the chief of the Apostles met their glorious end. For these are the men, through whom the light of Christ's gospel shone on thee, O Rome, and through whom thou, who wast the teacher of error, wast made the disciple of Truth.” Pope Leo the Great (regn. A.D. 440-461), Sermon LXXXII (ante A.D. 461)