Michelle - as others have pointed out, there are several successors to the Apostles named in the Bible itself. Others can be determined by chasing back the chain of bishops in the ancient city-states of the Roman Empire. Itās not entirely straightforward one-to-one because, of course, the Apostles for the most part travelled widely, proclaiming the gospel āto the ends of the earth,ā and either ordaining local leaders to the churches when they moved on, or sending trusted associates to lead them.
Here are a few of those with the cities they served:
Lucius was one of the converts at Pentecost, he was later ordained to the see of Cyrene by the Apostles. Another early convert, Quartus, was the first bishop of Beirut. Rufus, the son of Simon who was forced by the Romans to help Jesus carry the Cross, was later bishop of Thebes.
Peter ordained Evodius in Antioch, who was in turn succeeded by Ignatius.
Peter ordained Ancletus and Clement as co-leaders in Rome, after his death he was succeeded by Linus (Paulās assistant in Rome) and then by Ancletus and Clement (the order is not certain and they may well have served together).
Andrew ordained Stachys as bishop of Byzantium/Constantinople (now Istanbul) and Paremenas as bishop of Soli.
James son of Zebedee is supposed to have ordained Peter of Braga in Portugal.
John, son of Zebedee, ordained Polycarp bishop of Smyrna, he was succeeded by Papirius.
Philip ordained Narcissus as bishop of Athens and Papias in Phrygia (although Papias may have succeeded Stachys, who was ordained by Bartholomew aka Nathaniel, or they may have co-led that region).
Paul, of course, ordained a bishop to lead the church in every major city in which he evangelized, many of whom are named in Scripture - Timothy in Ephesus, (who was succeeded by Onesimus, possibly the same one named in Philemon). Paul ordained Titus and sent him to settle things in Crete. He ordained Aristarchus and Jason to servce in Apamea and Tarsus in Syria, Crescens in Galatia, Epaphroditus in Philippi, Clement in Sardis, and probably others whose names have been forgotten. Apollos may have served as bishop of Corinth.
The Mormon faith depends almost essentially on the belief that the true faith was lost within a generation of the death of the apostles and was restored by a vision to Joseph Smith, which is why they reject the Apostolic Succession. (Ironically, their doctrines have a lot in common with the gnostic heresies that were such a threat to the Church in the immediate post-apostolic era.)