Its story
During Diocletian's persecution (303–305), authorities required Christians to hand over their sacred books. Those who did were called traditores ('handers-over'). Around 311–312 Caecilian was consecrated bishop of Carthage; his opponents claimed that one of his consecrators, Felix of Aptunga, had been a traditor, so the consecration was void. They elected a rival bishop, Majorinus, soon succeeded by Donatus, after whom the movement was named.
The Donatists saw themselves as the Church of the martyrs, faithful to the African tradition of Cyprian of Carthage (3rd century), who had held that baptism given outside the Church was invalid. They therefore rebaptized those coming from the other communion. In much of North Africa they became the majority, and their protest also had social and regional overtones against Rome and imperial power; a radical wing, the 'Circumcellions', sometimes resorted to violence.
Constantine referred the case to bishops: a synod at Rome (313) and the Council of Arles (314) ruled for Caecilian, and Arles accepted baptism given by heretics in the name of the Trinity. Optatus of Milevis (c. 365) and then Augustine of Hippo developed the answer: Christ is the one who baptizes, and the sacrament is valid even if the minister is unworthy (what would later be called ex opere operato); the Church on earth is a mixed body of wheat and tares until the final judgment. In 411 a great conference at Carthage, chaired by the imperial commissioner Marcellinus, ruled against the Donatists, and repressive laws followed.
The controversy left a mixed legacy. On one hand it fixed the doctrine, shared by Catholics, Orthodox (with nuances) and most Protestants, that a sacrament's efficacy does not depend on the minister's personal holiness. On the other, Augustine came to justify state coercion of schismatics ('compel them to come in', Luke 14:23), an argument later used in the history of religious intolerance. Donatism survived in residual form until the Arab conquest of North Africa.