Irenaeus (130-202)'s Against Heresies documents early church tradition of the apostles Paul and Peter and the successive leadership of the church from the apostles to 12 popes ending with Pope Eleutherius (d. 189 AD)

An engraving of Irenaeus (130-202), Bishop of Lugdunum in Gaul (now Lyons, France)

The Apostles pass the church onto the next generation before they go the way of all men and die

Adriaen van de Venne's Fishers of Men. Oil on panel, 1614.

It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about. For if the apostles had known hidden mysteries, which they were in the habit of imparting to "the perfect" apart and privily from the rest, they would have delivered them especially to those to whom they were also committing the Churches themselves. For they were desirous that these men should be very perfect and blameless in all things, whom also they were leaving behind as their successors, delivering up their own place of government to these men; which men, if they discharged their functions honestly, would be a great boon [to the Church], but if they should fall away, the direst calamity.

The Church was founded and organized at Rome by Peter and Paul and has pre-eminent authority

Apostles Peter and Paul by El Greco

Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say, ] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its pre-eminent authority, that is, the faithful everywhere, inasmuch as the apostolical tradition has been preserved continuously by those [faithful men] who exist everywhere.

Tradition from the apostles to Pope Linus to Pope Anacletus to Pope Clement to Pope Evaristus to Pope Alexander to Pope Sixtus to Pope Telephorus to Pope Hyginus to Pope Pius to Pope Anicetus to Pope Sorer to Pope Eleutherius

This fresco (1481–82) by Pietro Perugino in the Sistine Chapel shows Jesus giving the keys of heaven to Saint Peter.

The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles, proclaiming the one God, omnipotent, the Maker of heaven and earth, the Creator of man, who brought on the deluge, and called Abraham, who led the people from the land of Egypt, spake with Moses, set forth the law, sent the prophets, and who has prepared fire for the devil and his angels. From this document, whosoever chooses to do so, may learn that He, the Father of our Lord Jesus Christ, was preached by the Churches, and may also understand the apostolical tradition of the Church, since this Epistle is of older date than these men who are now propagating falsehood, and who conjure into existence another god beyond the Creator and the Maker of all existing things. To this Clement there succeeded Evaristus. Alexander followed Evaristus; then, sixth from the apostles, Sixtus was appointed; after him, Telephorus, who was gloriously martyred; then Hyginus; after him, Pius; then after him, Anicetus. Sorer having succeeded Anicetus, Eleutherius does now, in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth.

Polycarp (80-167 AD) combated heretics including Valentinus, Marcion, and Cerinthus

S. Polycarpus, engraving by Michael Burghers, ca 1685

But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom, departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true. To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time,-a man who was of much greater weight, and a more steadfast witness of truth, than Valentinus, and Marcion, and the rest of the heretics. He it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles,-that, namely, which is handed down by the Church. There are also those who heard from him that John, the disciple of the Lord, going to bathe at Ephesus, and perceiving Cerinthus within, rushed out of the bath-house without bathing, exclaiming, "Let us fly, lest even the bath-house fall down, because Cerinthus, the enemy of the truth, is within." And Polycarp himself replied to Marcion, who met him on one occasion, and said, "Dost thou know me? ""I do know thee, the first-born of Satan." Such was the horror which the apostles and their disciples had against holding even verbal communication with any corrupters of the truth; as Paul also says, "A man that is an heretic, after the first and second admonition, reject; knowing that he that is such is subverted, and sinneth, being condemned of himself." There is also a very powerful Epistle of Polycarp written to the Philippians, from which those who choose to do so, and are anxious about their salvation, can learn the character of his faith, and the preaching of the truth. Then, again, the Church in Ephesus, founded by Paul, and having John remaining among them permanently until the times of Trajan, is a true witness of the tradition of the apostles.

Source: Irenaeus, Against Heresies, Book 3, Chapter 3.