The Fathers Speak: St. Irenaeus
St. Irenaeus was a second century bishop of Lyons, Gaul (modern-day France). He learned the Faith from St. Polycarp, who himself received it from St. John, the Beloved Apostle. Irenaeus’s most famous work was Against Heresies, in which he refutes the growing Gnostic heresies of his day. The following passage is taken from that work and explains Apostolic succession and the primacy of the bishop of Rome.
…pointing out here the successions of the bishops of the greatest and most ancient Church known to all, founded and organized at Rome by the two most glorious Apostles, Peter and Paul, that Church which has the tradition and the faith which comes down to us after having been announced to men by the Apostles. For with this Church, because of its superior origin, all Churches must agree, that is, all the faithful in the whole world; and it is in her that the faithful everywhere have maintained the Apostolic tradition.
The blessed Apostles, having founded and built up the Church [of Rome], they handed over the office of the episcopate to Linus. Paul makes mention of this Linus in the Epistle to Timothy. To him succeeded Anencletus; and after him, in the third place from the Apostles, Clement was chosen for the episcopate. He had seen the Apostles and was acquainted with them…To this Clement, Evaristus succeeded; and Alexander succeeded Evaristus. Then, sixth after the Apostles, Sixtus was appointed; after him, Telesphorus, who also was gloriously martyred. Then Hyginus; after him, Pius; and after him, Anicetus. Soter succeeded Anicetus, and now, in the twelfth place after the Apostles, the lot of the episcopate has fallen to Eleutherus. In this order, and by the teaching of the Apostles handed down in the Church, the preaching of the truth has come down to us.