The most important issue is the Donatist controversy beginning in 312. Essentially, some of the members of the church handed over Christian books and denied that they were Christians during various persecutions. Two questions arose: (1) whether individuals should be reinstated and (2) were sacraments performed by these questionable figures valid?
The most important question theologically coming out of the Donatist controversy was that of whether a priest had to be in a state of grace for a sacrament he performed to be valid. St Augustine decisively answered in the negative in the 4th century and that has continued to be Roman doctrine. Thus, for example, if the priest who married you was latter exposed as a pedophile, this does not invalidate the marriage.