Who was the next Pope after Peter?

After the death and burial of Peter, likely in 67 A.D., Linus became the Bishop of Rome, and thus Pope. According to St. Irenaeus writing in the 2nd century, he was appointed in advance,

After the Holy Apostles (Peter and Paul) had founded and set the Church in order, they gave over the exercise of the episcopal office to Linus. The same Linus is mentioned by St. Paul in his Epistle to Timothy. His successor was Anacletus. (Against Heresies, III.3.3)

Pope Linus’s pontificate concluded with his own martyrdom in 76 A.D. The Church commemorates his death on September 23.