While adult baptism was the norm at the end of the 4th century, sometimes there were exceptions. Moreover, as the Church increasingly encouraged earlier baptisms, so did the number of those needing to avail themselves of the possibility of reconciliation increase.
Links:
Painting of Saint Fabiola by Jean-Jacques Henner
Basil of Caesarea’s recommended penances for serious sins
Letter of Saint Jerome on Saint Fabiola
Ambrose’s Eulogy of Valentinian II – important in development of theology of Baptism of Desire
Ambrose on Penance – Book I and Book II
Apostolic Constitutions – Book VIII, Chapter 9 concerns penance
Check out the other great Catholic podcasts at the Starquest Production Network
To listen, just click on the link below:
#390 – A History of the Catholic Church – Penance
catholicunderthehood.com 
