Being apostolic would mean that they retained the teachings of the apostle. Even Today, with all of the manuscripts and historical evidence, it is still difficult to determine just how closely the teachings of the Apostolic Fathers resembled to or agreed with Jesus’ teachings. The objective of these men was undoubtedly altruistic (noble), seeking to protect or support (make known) a certain orthodox Christianity. They denounced idolatry and loose morals. They believed that Jesus is the divine Son of God and that he was resurrected. However, they were not able to hold back the growing wave of apostasy. Sadly, being honest, some of them contributed to it.
Between A.D. 95 and about 165, a number of works were written by men who had known the apostles and the apostolic doctrine; they are known as the Apostolic Fathers.
Ignatius of Antioch [c.35-50 – c. 98-117 AD] also known as Ignatius Theophorus was an early Christian writer and bishop of Antioch. While en route to Rome, where he met his martyrdom, Ignatius wrote a series of letters.
Papias was a Greek Apostolic Father, Bishop of Hierapolis, and author who lived c. 60 – c. 135 AD. He wrote the Exposition of the Sayings of the Lord in five books.
Polycarp was a Christian bishop of Smyrna. According to the Martyrdom of Polycarp, he died a martyr, bound and burned at the stake, then stabbed when the fire failed to consume his body.
Clement of Rome belongs to a group of early church leaders that have been known since the seventeenth century as the “Apostolic Fathers.”
The epistle was written in the name of the Roman Church to the Christian brotherhood at Corinth. The author was Clement, the Bishop of the Roman Christians, but he does not write in his own name.
There is much in-depth information in this article: The Synoptic Gospels in the Ancient Church: The Testimony to the Priority of the Gospel of Matthew. We have a brief introduction to papyrus from Tyndale Bible Dictionary. We have a lengthy apologetic article on Papias and the arguments from higher critics by F. David Farnell. This is followed by Papias' writings from two leading scholars on the Apostolic Fathers, Michael W. Holmes, and J. B. Lightfoot.