The Catholic Church is led by successors of the apostles. Every priest has been ordained by a bishop, who was ordained by a bishop, who was ordained by a bishop, etc. all the way back to the apostles.
Here's an early list of popes from Irenaeus written in about 180 AD. It is part of his treatise titled
Against Heresies, and it was meant to help battle the threat of gnosticism which was an issue during his lifetime.
1. 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;
2. 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
knownChurch 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 preeminent authority, that is, the faithful everywhere, inasmuch as the
tradition has been preserved continuously by those [faithfulmen] who exist everywhere.
3. 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, spoke 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
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.
Soter 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.
CHURCH FATHERS: Against Heresies, III.3 (St. Irenaeus)