Linus was the first successor of Peter.
In other words, Peter was the first bishop of Rome and Linus was the second. Third came Cletus and fourth came Clement of Rome. The line of bishops that went through Peter and Clement was always considered the sacred line that belonged to the Church founded by Jesus. Today, Benedict XVI is 266th in that same line of succession. (which makes him the 265th successor of Peter)
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Here are the first ten popes of the Church:
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St. Peter (32-67 A.D.)
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St. Linus (67-76 A.D.)
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St. Anacletus (Cletus)(76-88 A.D.)
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St. Clement I (88-97 A.D.)
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St. Evaristus (97-105 A.D.)
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St. Alexander I (105-115 A.D.)
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St. Sixtus I (115-125 A.D.) -- also called Xystus I
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St. Telesphorus (125-136 A.D.)
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St. Hyginus (136-140 A.D.)
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St. Pius I (140-155 A.D.)
Here is a LIST OF ALL POPES
Here is where you can get a beautiful
poster of all popes from Peter to Benedict for your wall. 
In the early centuries of the Church, Church Fathers like St. Augustine and St. Irenaeus would prove to heretics that the Catholic Church was God’s Church simply by demonstrating that the Church’s head bishop was a direct successor of Peter.
The early defenders of the Church would sometimes list all the popes back to Peter.
For instance, here is Bishop Irenaues of Lyons, way back in 189 A.D., listing the popes up to his time in his letter “Against Heresies”:
“The blessed apostles [Peter and Paul], 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 letter to Timothy. To him succeeded Anacletus, and after him, in the third place from the apostles, Clement was chosen for the episcopate. He had seen the blessed apostles and was acquainted with them. It might be said that he still heard the echoes of the preaching of the apostles and had their traditions before his eyes. And not only he, for there were many still remaining who had been instructed by the apostles.”
Augustine also compiled pope lists like this.
Jerome, who notes in his letter “Against the Luciferians” that Pope Steven was the 23rd successor of Peter, cried out that he would follow no leader but Christ and would join with nobody but he who possessed the Chair of Peter. The Chair of Peter is the seat or position of authority that all successors of the pope sit in.
Jerome, the early Church’s greatest authority on Scripture, wrote in 396 A.D.: "I follow no leader but Christ and join in communion with none but your blessedness [Pope Damasus I], that is, with the chair of Peter. I know that this is the rock on which the Church has been built. Whoever eats the Lamb outside this house is profane. Anyone who is not in the ark of Noah will perish when the flood prevails"
CATHOLIC TRIVIA