Apostolic succession in the Fathers

The unbroken transmission of the apostolic faith and ministry from the Twelve to the bishops of each church through ordination. From the earliest fathers — Clement of Rome, Ignatius, Irenaeus, Cyprian — succession in the chair (cathedra) of each apostolic see was the surety that the church there held the apostolic Tradition. The bishop is the centre of the church's unity, the celebrant of the one Eucharist, and the guarantor of the deposit of faith.

1Clem 42.1
First Epistle of Clement to the Corinthians · Saint Clement of Rome · Apostolic Fathers, tr. J. B. Lightfoot
The Apostles received the Gospel for us from the Lord Jesus Christ; Jesus Christ was sent forth from God.
St Clement of Rome (c. 96 AD), the earliest extant non-canonical writing on the question: "The Apostles received the Gospel for us from the Lord Jesus Christ... So then, both were duly appointed in the will of God. Having therefore received a charge... they went forth preaching... and appointed their firstfruits... to be bishops and deacons unto them that should believe." The chain Christ → Apostles → bishops as the order of the Church itself.
St Clement of Rome, First Epistle to the Corinthians 42
1Clem 44.1
First Epistle of Clement to the Corinthians · Saint Clement of Rome · Apostolic Fathers, tr. J. B. Lightfoot
And our Apostles knew through our Lord Jesus Christ that there would be strife over the name of the bishop's office.
St Clement continues: "And our Apostles knew through our Lord Jesus Christ that there would be strife over the name of the bishop's office. For this cause therefore, having received complete foreknowledge, they appointed the aforesaid persons, and afterwards they provided a continuance, that if these should fall asleep, other approved men should succeed to their ministration." The earliest explicit doctrine of episcopal succession.
St Clement of Rome, First Epistle to the Corinthians 44
Ign Magn 6.1
Ignatius to the Magnesians · Hieromartyr Ignatius the God-bearer of Antioch · Apostolic Fathers, tr. J. B. Lightfoot
Seeing then that in the aforementioned persons I beheld your whole people in faith and embraced them, I advise you, be ye zealous to do all things in godly concord, the bishop presiding after the likeness of God and the presbyters after the likeness of the council of the Apostles, with the deacons also who are most dear to me, having been entrusted with the diaconate of Jesus Christ, who was with the Father before the worlds and appeared at the end of time.
St Ignatius of Antioch: "Do nothing without the bishop and the presbyters... I exhort you, be zealous to do all things in godly concord, the bishop presiding after the likeness of God and the presbyters after the likeness of the council of the Apostles." The bishop is the icon of God the Father; the presbyterium is the icon of the Apostolic college.
St Ignatius, Epistle to the Magnesians 6
Ign Smyrn 8.1
Ignatius to the Smyrnaeans · Hieromartyr Ignatius the God-bearer of Antioch · Apostolic Fathers, tr. J. B. Lightfoot
[But] shun divisions, as the beginning of evils. Do ye all follow your bishop, as Jesus Christ followed the Father, and the presbytery as the Apostles; and to the deacons pay respect, as to God's commandment. Let no man do aught of things pertaining to the Church apart from the bishop. Let that be held a valid eucharist which is under the bishop or one to whom he shall have committed it.
St Ignatius: "Let no man do anything pertaining to the Church apart from the bishop. Let that be held a valid Eucharist which is under the bishop or one to whom he shall have committed it." The Eucharist and the bishop are inseparable; succession is not legal but liturgical.
St Ignatius, Epistle to the Smyrnaeans 8
AH 3.3.1
Against Heresies (Adversus Haereses) · Saint Irenaeus of Lyons · Ante-Nicene Fathers (Roberts–Donaldson)
1. When, however, they are confuted from the Scriptures, they turn round and accuse these same Scriptures, as if they were not correct, nor of authority, and [assert] that they are ambiguous, and that the truth cannot be extracted from them by those who are ignorant of tradition. For [they allege] that the truth was not delivered by means of written documents, but vivâ voce: wherefore also Paul declared, “But we speak wisdom among those that are perfect, but not the wisdom of this world.” And this wisdom each one of them alleges to be the fiction of his own inventing, forsooth; so that, according to their idea, the truth properly resides at one time in Valentinus, at another in Marcion, at another in Cerinthus, then afterwards in Basilides, or has even been indifferently in any other opponent, who could speak nothing pertaining to salvation. For every one of these men, being altogether of a perverse disposition, depraving the system of truth, is not ashamed to preach himself.
St Irenaeus (c. 180), against the Gnostics: "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 the succession of these men to our own times." Apostolic succession as the touchstone of authentic Tradition against secret Gnostic teaching.
St Irenaeus, Against Heresies III.3.1
AH 3.4.4
Against Heresies (Adversus Haereses) · Saint Irenaeus of Lyons · Ante-Nicene Fathers (Roberts–Donaldson)
4. 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 stedfast 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.
St Irenaeus on the church of Smyrna, where St Polycarp "had been instructed by Apostles, and had familiar intercourse with many who had seen Christ, and had also been appointed bishop by Apostles in Asia, in the Church of Smyrna." The chain runs back to the eyewitnesses of the Lord in a single human lifetime.
St Irenaeus, Against Heresies III.3.4
AH 4.26.2
Against Heresies (Adversus Haereses) · Saint Irenaeus of Lyons · Ante-Nicene Fathers (Roberts–Donaldson)
2. This fact is indeed set forth by many other [occurrences], but typically by [the history of] Thamar, Judah’s daughter-in-law. For when she had conceived twins, one of them put forth his hand first; and as the midwife supposed that he was the first-born, she bound a scarlet token on his hand. But after this had been done, and he had drawn back his hand, his brother Phares came forth the first; then, after him, Zara, upon whom was the scarlet line, [was born] the second: the Scripture clearly pointing out that people which possessed the scarlet sign, that is, faith in a state of circumcision, which was shown beforehand, indeed, in the patriarchs first; but after that withdrawn, that his brother might be born; and also, in like manner, him who was the elder, as being born in the second place, [him] who was distinguished by the scarlet token which was [fastened] on him, that is, the passion of the Just One, which was prefigured from the beginning in Abel, and described by the prophets, but perfected in the last times in the Son of God.
St Irenaeus: it is necessary to obey those presbyters who are in the Church — those who, as we have shown, possess the succession from the Apostles; those who, together with the succession of the episcopate, have received the certain gift of truth (charisma veritatis certum), according to the good pleasure of the Father.
St Irenaeus, Against Heresies IV.26.2
Praescr 32.1
The Prescription Against Heretics · Tertullian of Carthage · Ante-Nicene Fathers (Roberts–Donaldson)
But if there be any (heresies) which are bold enough to plant themselves in the midst of the apostolic age, that they may thereby seem to have been handed down by the apostles, because they existed in the time of the apostles, we can say: Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men,—a man, moreover, who continued stedfast with the apostles. For this is the manner in which the apostolic churches transmit their registers: as the church of Smyrna, which records that Polycarp was placed therein by John; as also the church of Rome, which makes Clement to have been ordained in like manner by Peter. In exactly the same way the other churches likewise exhibit (their several worthies), whom, as having been appointed to their episcopal places by apostles, they regard as transmitters of the apostolic seed. Let the heretics contrive something of the same kind. For after their blasphemy, what is there that is unlawful for them (to attempt)? But should they even effect the contrivance, they will not advance a step. For their very doctrine, after comparison with that of the apostles, will declare, by its own diversity and contrariety, that it had for its author neither an apostle nor an apostolic man; because, as the apostles would never have taught things which were self-contradictory, so the apostolic men would not have inculcated teaching different from the apostles, unless they who received their instruction from the apostles went and preached in a contrary manner. To this test, therefore will they be submitted for proof by those churches, who, although they derive not their founder from apostles or apostolic men (as being of much later date, for they are in fact being founded daily), yet, since they agree in the same faith, they are accounted as not less apostolic because they are akin in doctrine. Then let all the heresies, when challenged to these two tests by our apostolic church, offer their proof of how they deem themselves to be apostolic. But in truth they neither are so, nor are they able to prove themselves to be what they are not. Nor are they admitted to peaceful relations and communion by such churches as are in any way connected with apostles, inasmuch as they are in no sense themselves apostolic because of their diversity as to the mysteries of the faith.
Tertullian (c. 200): "Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that the first bishop of theirs shall be able to show for his ordainer and predecessor some one of the Apostles or of apostolic men." Apostolic succession as the legal proof of authentic Church identity.
Tertullian, Prescription Against Heretics 32
CypTr 1.1.5
Treatises of Cyprian · Saint Cyprian of Carthage · Ante-Nicene Fathers (Roberts–Donaldson)
Argument.—On the Occasion of the Schism of Novatian, to Keep Back from Him the Carthaginians, Who Already Were Not Averse to Him, on Account of Novatus and Some Other Presbyters of His Church, Who Had Originated the Whole Disturbance, Cyprian Wrote This Treatise. And First of All, Fortifying Them Against the Deceits of These, He Exhorts Them to Constancy, and Instructs Them that Heresies Exist Because Christ, the Head of the Church, is Not Looked To, that the Common Commission First Entrusted to Peter is Contemned, and the One Church and the One Episcopate are Deserted. Then He Proves, as Well by the Scriptures as by the Figures of the Old and New Testament, the Unity of the Church.
St Cyprian of Carthage in On the Unity of the Church: "The episcopate is one, of which a part is held by each without division of the whole. The Church also is one, which is spread abroad far and wide into a multitude by an increase of fruitfulness." Each bishop holds the whole episcopate, not a fragment of it — collegial unity without subordination.
St Cyprian, On the Unity of the Catholic Church 5
CypEp 26.1
Epistles of Cyprian · Saint Cyprian of Carthage · Ante-Nicene Fathers (Roberts–Donaldson)
Epistle XXVI.
St Cyprian: "You ought to know that the bishop is in the Church, and the Church in the bishop; and if any one be not with the bishop, that he is not in the Church." The Church is constituted by the gathering around the bishop in succession from the Apostles — outside this, sacramental life is not possible.
St Cyprian, Epistle 66 (al. 69) to Florentius Pupianus