There were seven councils held at Carthage between 253 and 256. 

When the Pagan Roman Emperor Decius died in 250 AD the persecution of Christians stopped. At that time the question was raised what to do with those who denied Christ and sacrificed to pagan gods in order not to be put to death. In the past common practice was to make a public confession, do some penance then after a time they would be received back into the church.

Novatian (210-280 AD) who is famous for his work on the Trinity in 257, began to teach that no one should be allowed back into the church after having denied Christ in the persecution. Those that followed this teaching were supposedly called Cathari, or pure ones. Later in church history a much different group will use the name Cathari. Some have tried to use Novatian and the pure ones as a basis in history for the puritans. Claiming they were Anabaptists fighting against infant baptism but as we will see this is not the case.

Others said people should only be let back into the church if they would be rebaptized. 

Cyprian (200-258AD) pupil of Tertullian, was definitely a sacramentalist. He also believed in a literal Antichrist. They baptize in the name of the Father, son, and the Holy Spirit. He stated you can't be saved with out baptism as it is for the remission of sins.

7th council Anyone baptized by a heretic (that has not been baptized in the church) must be rebaptized. Heretics are defined as non-Trinitarians; Marcion is given as an example. If one has been baptized as a Christian then joins a cult and repents and comes back into the church, he is not rebaptized, but makes a public confession and may do some penance then after a time is received back into the church.

If the church will not baptize a non-Trinitarians, it is simply making a public statement that it holds non-Trinitarians to be non-Christians, by this time baptism is look at as a means to salvation. Until this time the church taught the sacraments were symbolic only. Note: Novatian will later be called a heretic because of his views on church policy rather then on the doctrine of the trinity.

From the writings of Cyprian:

• Epistle 75 - Giving baptism by sprinkling to someone on their deathbed who can't come to church and be baptized is acceptable.
• 3rd council Epistle 58 - They had been baptizing infants for a while. The debate was if one had to wait till the infant was 8 days old or not.
• 5th council Epistle 69 - The priest giving the blessing to make water holy water must be in good standing with the catholic church (not Roman Catholic) otherwise the water is just water and the baptism is null and void.
• Epistle 70 - His colleague Stephen (Pope of Rome) is presumptuous.
• 6th council Epistle 71 - Baptism by dipping is the standard. Each prelate rules his own churches and other bishops do not interfere. Bishop Stephen of Rome is just another colleague.
• Epistle 72 - Marcion baptized in the name of Jesus only and not in the triune formula.
• Unity of the Church 4 - Peter is equal to the other apostles in the authority of binding and loosing, and forgiving sins. Whosoever does not hold to this teaching of unity "strives against and resists the church" and does not hold to "true faith"
• Unity of the Church 10 - Those who "appoint themselves as prelates with out any law of ordination" are the chaff separated from the wheat. (compare to Papal Infallibility)

Miscellaneous notes from the Epistles of Cyprian:

• 21 - Lucian believes in a literal antichrist.
• 30 - Mentions a heaven and hell, no hint of purgatory.
• 39 - Quotes Susanna.
• 40 - Cyprian rebukes Rome for ordaining Novatian (unlawful ordination and in opposition to the Catholic Church) and will not recognize or be in communion with any who are a part of his faction.
• 43 - Cyprian exhorts the Roman confessors to return to the Unity of the Church.
• 45 - Cornelius bishop of Rome writes to Cyprian telling him that the confessors have returned to unity.
• 48 - Rome should follow the precedence of Carthage. Since Carthage excommunicated Novatian, Rome should likewise follow. Lists all the sins of Novatian and his followers.
• 51-  After the persecution all the remaining Carthaginian bishops got together to debate what to do with the lapsed, both side appealing to scripture and the only authority. Then notified Rome of their decision. Rome had a similar council and came to the same conclusion. No confession or repentance can be made after death. (compare to the doctrine of Purgatory)
• 54 - The laity (secular government) decide what to do with evil people; heretics that murder, rape, and steal ect.) It does not matter how large a number depart from the unity of the early church true Christians remain faithful.
• 62 - There were some who were using water instead of wine mixed with water for communion (remnants of the Encratites, also referred to as Hydro-parastatae) He uses examples of Noah, Melchizedek of the use of wine. He quotes Paul in "even if an angel preaches another gospel" and says using water is perverse. Note 8: The heathen detected Christians by the smell of wine when looking for victims for the persecutor.
• 63 - Clerics should not get mixed up in secular affairs.
• 67 - Stephen was deceived by Basilides and Novatian. Cyprian states the prophecies of the Antichrist fore told of heresies rising up in the church at the end of the world.
• 73 - Cyprian wrote to bishop Stephen of Rome because he was in "error" by forbidding anyone coming from a heresy to be baptized in the church. Judging the baptism of heretics to be lawful. Stephen may have considered heretics to be only non-Trinitarians and therefore not consider Novatian a heretic. Stephen said that since the heretics do not rebaptize a convert who came to them from us, we should not rebaptize one of them who converts to the true church. Cyprian appears to believe any heretic must be rebaptized to become a Christian.
• 74 - Cataphrygians, and their new prophecies have a false Christ that of Montanus. Some heretics state there is no Holy Spirit. Point how can baptism and the new birth of the holy spirit be given if the baptizer does not believe in a Holy Spirit. Cyprian does not accept baptism from those who follow the “new prophets” of the Montantists even if the believe in the Trinity.  If a catechumen dies before baptism he is still saved. Example of rebaptism is John's disciples who had not heard there is a Holy Spirit.
• 75 - Sprinkling on a deathbed and immersion in church are both practiced. The schismatic tries to "establish a throne for themselves and assume primacy and the right of baptism"
• 82 - One does not voluntarily offer himself as a martyr.